"nurse (n.1)
12c., nurrice "wet nurse, foster-mother to a young child," from O.Fr. norrice (source of proper name Norris), from L.L. *nutricia "nurse, governess, tutoress," from L., fem. of nutricius "that suckles, nourishes," from nutrix (gen. nutricis) "wet nurse," from nutrire "to suckle" (see nourish). Meaning "person who takes care of sick" first recorded in Eng. 1590; the verb is first attested 1535 in sense of "to suckle (an infant)," 1526 in the passive sense, alt. of M.E. nurshen (13c.; see nourish), originally "to bring up or suckle a child," sense of "take care of (a sick person)" is first recorded 1736."

It seems obvious that the origin of nurse does not directly refer to breast feeding, elsewise it would not be necessary to refer to 'wet nursing'. From the above, and for that matter from what Sharon A posted, it appears that 'nursing' has always meant the care of another, whether in providing food or assisting in health.

I think we would agree that Clara Barton did not suckle the soldiers in her care. (Joke, OK?)