According to Wikipedia, the study of medicine at Salerno goes back to Roman times
It does; my point is that they were granting medical degrees in a program that is very similar to modern M.D.
We have, by the way, not only the works of people like Galen, but a fair number of medieval English, Irish, and Welsh medical texts.
The Welsh in particular had specific families known from the middle ages for the practice of medicine, with family libraries of mss.
Obviously, if you're writing fantasy, you decide how medicine, and practioners thereof, work in your book.