Avodah Zarah 28 - Going to an Idol-Worshipping Doctor
It is assumed that a pagan doctor will try to kill a Jewish patient if he can. However, there are gradations. He will not hesitate to let a critically ill Jewish patient die, since this will not harm his reputation. He may also kill a healthy patient, since other pagans will understand that he did it out of malice, not incompetence. Accordingly, one with a non life-threatening illness may go to a pagan doctor.
Rabbi Yochanan suffered from a sickness called tzafdina, a tooth ailment. He went to a certain idolatrous noblewoman for treatment. She made him a remedy on Thursday and Friday, but on Saturday he could not come, because of many students, and asked for a recipe. She said, "Swear that you will not reveal it to anyone!" He said, "To the God of Israel I will not reveal" (but to the people I will), then taught it in public. But he swore?!? - He explained to her the trick right away. Some say that she committed suicide, but some say that she converted, impressed with Rabbi Yochanan's selflessness.
Now, seeing that Rabbi Yochanan was ready to take the medicine on Saturday, it must have been a life-threatening disease, and if so, how was he allowed to go to an idolatrous doctor? - Rabbi Yochanan was a distinguished person whom the doctor would be afraid to harm.
Art: Edouard (Jean-Edouard) Vuillard - Doctor Viau in his Surgery