Can a patient ask a doctor out on a date?

Yes but if the patient is a patient of that doctor, the doctor would HAVE to refuse.

Doctors, like everyone else, have a rite to a private life and can go out on dates. However, the rules (for the doctor) are that the the relationship between a doctor and that doctor's patient must be professional, because doctors are in a position to take advantage of their patients (who may be feeling vulnerable).