I suppose it's more accurate to say that a doctor is quitting your employ, rather than firing you, but it's an interesting topic. A piece in today's NYT raises the issue from the professional ethics standpoint. According to AMA guidelines,
a doctor may withdraw from a case only if the doctor notifies a patient, the patient's relatives or responsible friends with enough advance notice for the patient to secure another physician. A physician, under the guidelines, can decline to treat a patient who requests a treatment that is known to be scientifically invalid or that is incompatible with the physician's personal, religious or moral beliefs.
Ethics, though, aren't the whole story. The law has something to say on the issue as well. In particular, states are considering laws that allow doctors, pharmacists, and other health professionals to refuse to perform certain procedures, dispense certain drugs, or otherwise participate in activities to which they have a moral or religious objection. The AMA's guidelines recognize that physicians may have religious objections to certain treatments, but AMA guidelines don't dictate the legal consequences of acting on such an objection. In Wisconsin, the state Senate is considering a bill that provides exemptions from civil liability, professional discipline, and adverse employment consequences to health care professionals and health care facilities that refuse to participate in abortions, sterilizations, and a half-dozen other activities. Link to a story about the bill in the UW-Madison Daily Cardinal.