What is the proof that nature is deterministic?

I assume you meant not on the quantum level but on the level of belief,
so there isn’t any, and there won’t be.
At most he can argue that we see around us that nature is deterministic, and the burden of proof is on the one who wants to claim otherwise.
But how much testimony is there about miracles? Throughout all of history, and supernatural things too—there’s no shortage of evidence.

There is no certain proof for anything in the world. If you don’t accept the laws of physics, you’re welcome to live in a fantasy. We draw conclusions from observing the world in every aspect of our lives, and you can always wonder whether we’re right. It’s worth recalling the Ran and Maimonides that I brought at the beginning of the column.
How did Maimonides know that there are no demons? How did he know there were separate intellects and that the world is made up of four elements? That was the science of his time (which of course was far less well founded than ours). Not to mention: how did the Sages know all kinds of scientific and other information that they wrote?

The Rabbi of course ignored the fact that there are countless testimonies all over the world to supernatural things. It makes no sense not to accept such a mass of testimony throughout all of history and across countless groups of people.
I agree that there is also “nature,” but not only nature.
And Rabbi Eliezer Papo already resolved this issue at length in his book Pele Yoetz. The Rabbi could have included him in the trilogy that has not yet been published, by means of a few interpretive reinterpretations, where most people are in the third section.
“And here inquiry has found a place to rest: since God kills and gives life, what can the doctor give and what can he add? If it has been decreed that a person will die, then even if all the doctors in the world stand by him, they will not be able to save his life from death. About this they said: the doctor’s mistake is the Creator’s will. And none can save from His hand. And if it has been decreed that he will live even without a doctor, the Holy One, blessed be He, brings about circumstances and brings him his healing, for the Omnipresent has many ways and with Him is abundant redemption, and He will send forth His word and heal them.
But the answer to this, and to many such inquiries, is that there are three kinds of decrees. If he is a man whose merit suspends judgment, and his merit is strong and his fortune healthy, then even if he has no doctor and does things contrary to his illness, he will not die but will surely live.
And there are those for whom death has been decreed, God have mercy—if he guards himself with every safeguard in the world, the watchman keeps vigil in vain, and the doctors will not help and will not save.
But there are those who, because of their sins, are left under nature: if he guards himself and follows remedies according to nature, he will surely live, and if not, he will die. About such a case it is said: ‘Sometimes a person perishes without justice’ (Proverbs 13:23).”

By the way, it would be appropriate to connect the question here to what they started discussing here:
https://mikyab.net/%D7%A9%D7%95%D7%AA/%d7%aa%d7%92%d7%95%d7%91%d7%94-%d7%9C%d7%A8%D7%98#comment-26224

I’m of course not ignoring it. You’re ignoring the number of fools walking around the world. And even if there is this or that testimony, at most it proves a particular case (as I said, I cannot rule out sporadic involvement).
Very nice of you to agree that there is also nature. 🙂