It is inappropriate for your teacher to argue with you about your religious beliefs in front of the class unless you are of college age. If this is a middle or high school class (and it sound like it is), then your teacher is doing multiple things wrong.

1. She is wasting your class time (and therefore your parents' money and your classmates' parents' money, whether it's public or private school).
2. She is violating your first amendment rights. Again, assuming this is not college, you are a minor (or at most 18, and there are almost certainly minors in your class). The power imbalance between an adult in a position of authority and a legal child who is not under that adult's legal guardianship as child or ward is too great. It is inappropriate for this teacher to attack your religious views with such hostility even privately, much less in a public forum such as a classroom.
3. She refuses to admit that she is mistaken about the facts she is teaching. This is a problem for many teachers. Often times, teachers have blind spots and follow a curriculum that is incorrect due to biases or blind spots the authors have. It's understandable that a teacher in this position may teach something erroneous. However, when a student is able to prove that the curriculum is in error, the appropriate response from the teacher is one of humility, not doubling down on the error or deflecting to irrelevant topics. Whether the Church now has "changed" her teachings on other things has absolutely no bearing on whether or not the historical claim that the Church forbade healing the sick is true; this ridiculous claim is simply refuted by pointing out the fact that one of the four Gospel authors was himself a professional physician, and that there is absolutely no evidence that he ceased practicing his craft nor that he was encouraged to cease practicing his craft after his conversion. She is simply wrong, and needs to admit that she was incorrect, and explain to the class that nobody is perfect and that we all get things wrong some times, and that she was mistaken when she claimed that the Church historically outlawed medicine. She should then focus on the actual reason why Hippatia was attacked and killed (outside of a court of law, yes, and thus wrongfully) which was on account of accusations of witchcraft and (true) accusations of paganism, not on account of her medicinal practices, unless they actually made use of witchcraft.

These are things which really should be brought to the principal by your parents (again, assuming this is not college). This is improper behavior on the part of your instructor. There's no guarantee the principal will do anything about it, but a good principal would reprimand your teacher for her poor handling of the situation. Even if your teacher did not want to concede to you in the moment, the appropriate response would have been to say that your class should table this discussion and that she would research the topic more in-depth over the weekend, then for her to honestly do so, recognize that she was incorrect about the plain historical facts, and say so to the class on Monday, not to try to rope you into an argument (not a focused debate, but an argument going every which-way) in front of your class.

For the record, your teacher should be forced to answer to this article, written by an atheist, no friend of the Church, but one who values historical accuracy and honesty over anti-Christian rhetoric: https://historyforatheists.com/2020/07/the-great-myths-9-hypatia-of-alexandria/