health care vs Medicare

health care and Medicare both are nouns.

Word	Noun	Adjective	Verb	Adverb
health care	Yes	No	No	No
Medicare	Yes	No	No	No

As nouns, Medicare is a hyponym of health care; that is, Medicare is a word with a more specific, narrower meaning than health care:

health care: social insurance for the ill and injured
Medicare: health care for the aged; a federally administered system of health insurance available to persons aged 65 and over

Other hyponyms of health care include Medicaid, primary health care.

health care (noun)	Medicare (noun)
social insurance for the ill and injured	health care for the aged; a federally administered system of health insurance available to persons aged 65 and over
the preservation of mental and physical health by preventing or treating illness through services offered by the health profession