(1) Universal health care isn’t socialism, nor is it socialized medicine. Socialism involves collective ownership of the means of production. Socialized medicine is when the government owns the hospitals and employs the health care workers.
(2) It is not difficult to make a case for universal health care in capitalist economic terms, and any rational supporter of universal health care should be interested in making it regardless of whether or not they are capitalists. Why? Because the people they still need to convince are capitalists, and it’s easier to convince someone on their own terms than to rebuild their worldview from the ground up.
(3) As far as your question title is concerned, socialism and capitalism are incompatible precisely because they are diametrically opposed on the issue of who should own the means of production. This does not mean, however, that you can’t have a society in which some things are more like they would be under a purely socialist system and other things are more like they would be under a purely capitalist system (see mixed economy).