late 14c., "spiritual guide, ecclesiastic responsible for the spiritual welfare of those in his charge; parish priest," from Medieval Latin curatus "one responsible for the care (of souls)," from Latin curatus, past participle of curare "to take care of" (see cure (v.)). Church of England sense of "paid deputy priest of a parish" first recorded 1550s.
"be in charge of, manage" a museum, gallery, art exhibit, etc., by 1979 (implied in curated), a back-formation from curator or curation. Related: Curating. An earlier verb, curatize (1801) meant "be a (church) curate."