Answer

Hello Or.
In my opinion, the fact that a certain passage is included in the Talmud does not automatically make it Torah. There are parts of the Talmud whose study is a clear waste of Torah study. To be sure, this is also my view regarding all aggadic literature, unless one succeeds in finding in it added spiritual and Torah value (which in my opinion does not happen), in which case it becomes Torah on the personal level. But a passage that teaches something about medicine is simply a waste of Torah study. Therefore the question is not really about the Sabbath. On the Sabbath one may read enjoyable literary passages (for whoever enjoys that).
However, in quite a few cases medicine is mentioned in a halakhic context (such as “the remedy is in our hands” in the passage about one who is seized, and the like), and there there is Torah value to the study, while the medical information is only the background through which the Torah information is conveyed. When they tell me that one can cure cordiacus by eating crushed cow tail, and therefore cordiacus is not a complete lunatic, then even if today I think cow tail is of no use, I still learned that if something can be cured, then it is not considered insanity in Jewish law.
As a side note, all this is said on the assumption that there is a prohibition against studying medicine on the Sabbath. As is well known, regarding the study of the sciences, the medieval authorities (Rishonim) disagreed (Rashba and Maimonides; see Shulchan Arukh, sec. 307:17), but in medicine there is value beyond ordinary science, and therefore I do not see what prohibition there would be in it. And even from the standpoint of preparation on the Sabbath for a weekday, one should distinguish cases where the study has value in itself. And if it is for intellectual interest, that is certainly a good thing on the Sabbath. In any case, the custom today is certainly to be lenient about this, and in my opinion not only to be lenient but actually to recommend it.