cure (n.1)
c.1300, "care, heed," from Latin cura "care, concern, trouble," with many figurative extensions, such as "study; administration; a mistress," and also "means of healing, remedy," from Old Latin coira-, from PIE root *kois- "be concerned." Meaning "medical care" is late 14c.
cure (n.2)
parish priest, from French curé (13c.), from Medieval Latin curatus (see curate).
cure (v.)
late 14c., from Old French curer, from Latin curare "take care of," hence, in medical language, "treat medically, cure" (see cure (n.)). In reference to fish, pork, etc., first recorded 1743. Related: Cured; curing.

Most words for "cure, heal" in European languages originally applied to the person being treated but now can be used with reference to the disease, too. Relatively few show an ancient connection to words for "physician;" typically they are connected instead to words for "make whole" or "tend to" or even "conjurer." French guérir (with Italian guarir, Old Spanish guarir) is from a Germanic verb stem also found in in Gothic warjan, Old English wearian "ward off, prevent, defend" (see warrant (n.)).