Why exactly is this forbidden?
What is the reasoning here?

See the link I sent. There is an overview there.

I looked, but it says there as follows:
Two important halakhic decisors held that, from the standpoint of Jewish law itself and strictly speaking, there is no prohibition against artificial insemination of a married woman using the semen of a non-Jew, and in a time of great need, when the parents are suffering greatly because they long for a child, one may permit the use of a non-Jew’s semen. For, as stated, it is clear that the decree lest a brother marry his sister does not apply here. And as for the concerns that perhaps he will perform levirate marriage with his maternal brother’s wife and send his mother out to the marketplace, in their view this is not a basis for prohibition, because “we cannot add to the decrees of the Sages, and we do not find that they were concerned about this except in a case where this is in fact the reality, that it is unknown from whom the child came. But in a case like this, where the woman knows, and she is a trustworthy woman, from where do we have authority to innovate on our own a decree out of concern that perhaps she will be embarrassed to reveal the matter, and also will not withstand the test and will marry outside without halitzah? The rule is that regarding decrees of the Sages, we have only what they explicitly decreed, and we may not add decrees not mentioned in the Talmud.”

Meaning, I’m trying to understand on what basis you think this is forbidden. For the reasons mentioned above? Or for some other reason?

I studied the article.
And those who forbid really did not succeed in establishing a prohibition on something permitted.

And this is according to classic approaches like Rabbi Shlomo Zalman Auerbach and Rabbi Moshe Feinstein.

Certainly according to approaches like Rabbi Michael Abraham’s, that after the Sanhedrin there is no authority to invent new prohibitions.

I wonder where the Rabbi gets the authority, and on what basis, to forbid what is permitted?

I was talking about a sperm donation from a Jew, not from a non-Jew. With a non-Jew, that is a different discussion.
As for a Jew, in my opinion the main problem is not mentioned there, and that is creating mamzerim. In my opinion it is plausible that a child born this way is a mamzer (even though a dispute about this is cited there), and it is plausible that there is a prohibition on creating mamzerim. As for the source of this prohibition beyond the status of mamzer itself, it comes from the verse cited there about giving one’s seed to another man’s wife.
True, there are approaches that understand mamzer status as a punishment for forbidden intercourse, meaning for the prohibition violated by the parents, but in my opinion that may be true regarding other forbidden sexual relations. Regarding a married woman, there is an additional dimension in the very fact that he is not her husband. Something like this was written by Pnei Yehoshua regarding the applicability of kiddushin to a married woman—that beyond the sexual prohibition, she is another man’s wife—and I cited his words in my article on the nature of the guilt-offering.
It is true that some of the reasons mentioned there are slippery-slope arguments, which I oppose in principle for two reasons: 1. If there is not a very good reason, it is not proper to use them. But here maybe there is. 2. Because there is no authority to create a prohibition even if there are very good reasons.

And what is your opinion regarding sperm donation from a non-Jew?

With a non-Jew, the only difference is according to the view of Rabbenu Tam that intercourse with a non-Jew does not render her forbidden. Again, by reasoning this still does not seem right. One is still creating a son who is not the husband’s.
The commandment of kiddushin, in the accepted understanding, is what permits procreation, and here they are doing procreation without kiddushin.

But one can also fulfill procreation with an unmarried woman, and there you wrote that there is no problem with sperm donation even though it is done without kiddushin.

As for what you said that intercourse with a non-Jew does not render her forbidden, what did you mean by the words “does not render her forbidden”? After all, in the Shulchan Arukh, Even HaEzer (section 4, paragraph 19), it is stated that if a non-Jew has relations with a married woman, the child is not a mamzer.

A woman is obligated only in “He created it to be inhabited,” not in procreation. The owner of the sperm fulfills procreation this way as well, and my assumption is that he is the father. That is exactly the problem with such fertilization.
Therefore, the fact that procreation is fulfilled does not contradict there being a prohibition, since this creates a child from another father, one who is not her husband. What I wrote—that the Torah commands kiddushin as an instrument that permits the commandment of procreation—was not meant to say that without kiddushin one does not fulfill this commandment, but rather that the Torah wants us to perform kiddushin before procreation, or wants us to carry out procreation within a framework of kiddushin. Therefore when one does this outside a framework of kiddushin, one is not acting in accordance with the will of the Torah, and that is a prohibition. I did not say that one is nullifying procreation. That is not correct.
As for intercourse with a non-Jew: that is true, but there is a dispute among the medieval authorities whether intercourse with a non-Jew renders the woman forbidden to her husband and to the adulterer. Rabbenu Tam holds that it does not, and therefore in his view there is no forbidden intercourse here at all (“their flesh is the flesh of donkeys, and their emission is the emission of horses”). According to other views, such as Ri of Dampierre, intercourse with a non-Jew does render her forbidden, and therefore it is forbidden intercourse. Even so, he too would probably agree that the child is not a mamzer. It appears that the ruling in the Shulchan Arukh does not depend on this dispute among the medieval authorities, though see Pitchei Teshuvah, section 4, subsection 5, letter 1.
It is true that if the reasoning for prohibiting fertilization is the creation of a mamzer child, then here, with fertilization from a non-Jew, that prohibition does not apply. But if the reasoning is creating a child who is not from the husband, that applies here too.

Regarding what you wrote here:
“When this is done (procreation) outside a framework of kiddushin, one is not acting in accordance with the will of the Torah, and that is a prohibition”
What prohibition are we talking about? If you mean the prohibition of creating a child who is not from the husband—what is the source that such a thing is forbidden?

The prohibition is the nullification of the positive commandment of kiddushin. The Torah wants procreation to take place within a family framework, and in fertilization the procreation is not done that way.
This is true even according to the Rosh, who does not hold that kiddushin is a commandment. It is commonly thought that this is his dispute with Maimonides; in my opinion that is not necessarily so. Still, the Torah wants the man to take the woman before he has children with her.

But to the best of my knowledge, nullifying the positive commandment of kiddushin refers to intercourse without kiddushin, not to having children without kiddushin.

According to your view, does someone who lives with a woman in an ongoing intimate relationship without having children not nullify the positive commandment of kiddushin?

And besides, a non-Jew is not commanded in the positive commandment of kiddushin, and the woman is not either, so why should it be forbidden for a woman to receive sperm donation from a non-Jew?

I explained that the commandment of kiddushin is a kind of permit. That is, the Torah expects us to perform the commandment of procreation in such a way that an act of kiddushin precedes it. From this you can understand that if someone has children without kiddushin, he nullifies this commandment.
The halakhic decisors explained that it is not a permit for intercourse alone, but also, and primarily, for procreation. That is why I wrote that in my opinion even bringing children into the world without intercourse is a nullification of that commandment.
Someone who lives with a woman and does not have children is nullifying the commandment of procreation, not the commandment of kiddushin. After all, he did effect kiddushin, so how could you say that he nullified the commandment of kiddushin?
Even with a non-Jew, the woman is bringing children into the world without kiddushin having preceded it. The child is not born from a married couple, and therefore there is here a nullification of the commandment of kiddushin.

From where do you know that the woman is commanded in the positive commandment of kiddushin? After all, it says, “when a man takes a woman”—which implies that only the man is commanded to effect kiddushin and the woman is not commanded to become betrothed.

That is a valid claim. Even so, on the non-formal halakhic plane, it is still clear that the Torah wants children to come from a married couple. So by reasoning it seems that even the commandment of “He created it to be inhabited” is conditioned on kiddushin. But you are right that formally there is room to be lenient in the case of a woman, when we are talking about fertilization from a non-Jew.

By the way, “when a man takes a woman” only describes the action: that the man takes the woman. But that does not necessarily mean that the commandment is incumbent on him alone. It may be a commandment on both of them. It is likely that if people claim the commandment is on him, that is because only he is commanded in procreation, and kiddushin is what makes the commandment of procreation possible. It seems to me that if you look into it, you will see that the halakhic decisors do not explicitly say that the commandment is only on the man.

Regarding what you wrote, that the halakhic decisors wrote that the commandment of kiddushin is what permits procreation, I think that is because in their time there was no possibility of in vitro fertilization or sperm donation. Meaning, they meant to say that since in order to fulfill procreation one had to have intercourse, and since intercourse without kiddushin constitutes nullifying a positive commandment, therefore the commandment of kiddushin serves as a permit for both. But nowadays, when one can fulfill procreation without intercourse, what is your source that in such a case there is nullification of the positive commandment of kiddushin?

Also, in the verse it says, “when a man takes a woman and comes to her,” implying that the Torah connects kiddushin to intercourse as a permit, not kiddushin to procreation as a permit.

It is exactly the opposite. Statements that see kiddushin as a permit for intercourse are because in the past procreation was done only through intercourse. But by simple reasoning it is clear that it is a permit for procreation. Also because there is a commandment in procreation, not in intercourse, and that is also what all the halakhic decisors write. Seeing kiddushin as a permit for intercourse is what requires explanation.

I just now saw a responsum by Rabbi Cherlow permitting sperm donation even from a Jew:
https://www.ypt.co.il/6527

And he does not mention at all this issue of nullifying the positive commandment of kiddushin. How do you explain that?

I explain my own positions, not those of others. My argument also did not appear in Steinberg’s overview, which included only prohibiting opinions regarding donation from a Jew.

And I still can’t reconcile how you wrote in the original answer that it is permitted for an unmarried woman to become pregnant from sperm donation if there is nullification of the positive commandment of kiddushin here.

You are right. In light of the reasoning of nullifying the positive commandment of kiddushin, there is room to forbid it even for an unmarried woman, not only for a married one.

Suppose we are talking about a 40-year-old woman who no longer has time to find a normal husband within the time she has left to remain fertile, and therefore she wants a sperm donation. Does such a woman need to give this up and remain without children because of nullifying the positive commandment of kiddushin? Is it not preferable that she have a child and nullify that positive commandment so that the child can fulfill many commandments? In the spirit of “desecrate one Sabbath for him so that he may keep many Sabbaths.”

And I also wanted to ask, unrelatedly: could the problems mentioned regarding sperm donation to a married woman be solved if the woman divorces for the purpose of the donation, and after becoming pregnant returns to her husband?

I think so. There is perhaps nullification of a positive commandment here, if I am right, as against distress and the commandment of “He created it to be inhabited.”
It seems to me that yes.

“He created it to be inhabited” is rabbinic—how can it override a Torah-level positive commandment?

1. There is also the distress.
2. “He created it to be inhabited” is not really just rabbinic. It is regarded as a very fundamental obligation anchored in that verse as scriptural support.
3. For the parents, the child is also a need: a healing factor.
4. If localized human dignity overrides the nullification of a positive commandment, then all the more so childlessness should override it.

What does it mean that she doesn’t have time to find a normal husband? If she wants children that much (assuming that is what really motivates her and not the obligation of “He created it to be inhabited”), why shouldn’t she “compromise” on a husband for that sake?
It’s a bit like saying that someone doesn’t have time to find a normal job in order to buy tefillin, and since this causes him emotional distress—not because he wants to fulfill the positive commandment, but simply for psychological reasons—we would permit him to steal tefillin (from someone who is particular about this and therefore has not given permission) so he can put them on and feel good (and along the way fulfill the positive commandment), instead of working at a “degrading” or non-normal job.

*instead of working at a job

Sometimes there is no one with whom to compromise. Compromise is at times a very difficult demand. We are talking about a whole lifetime with someone she does not want. The Talmud in Ketubot 33 says that had Hananiah, Mishael, and Azariah been flogged, they would have worshipped the idol. A person can give himself over to death in a single moment, but not live an entire life of suffering.