Medicare (n.)

"pertaining or relating to the art or profession of healing or those who practice it," 1640s, from French médical, from Late Latin medicalis "of a physician," from Latin medicus "physician, surgeon, medical man" (n.); "healing, medicinal" (adj.), from medeor "to cure, heal," originally "know the best course for," from an early specialization of PIE root *med- "take appropriate measures" (source also of Avestan vi-mad- "physician"). "The meaning of medeor is based on a semantic shift from 'measure' to 'distribute a cure, heal'" [de Vaan]. The earlier adjective in English in this sense was medicinal. Related: Medically.

Old English caru, cearu "sorrow, anxiety, grief," also "burdens of mind; serious mental attention," in late Old English also "concern, anxiety caused by apprehension of evil or the weight of many burdens," from Proto-Germanic *karō "lament; grief, care" (source also of Old Saxon kara "sorrow;" Old High German chara "wail, lament;" Gothic kara "sorrow, trouble, care;" German Karfreitag "Good Friday;" see care (v.)).

The meaning "charge, oversight, attention or heed with a view to safety or protection" is attested from c. 1400; this is the sense in care of in addressing (1840). The meaning "object or matter of concern" is from 1580s. To take care of "take in hand, do" is from 1580s; take care "be careful" also is from 1580s.

The primary sense is that of inward grief, and the word is not connected, either in sense or form, with L. cura, care, of which the primary sense is pains or trouble bestowed upon something. [Century Dictionary]