Many Americans are really keen on "personal responsibility" and think the state has no business providing healthcare. They think everyone should pay their own way.
Of course they don't mind pooling resources to protect against huge loss for things like car insurance, but someone getting sick? Damn those failures, they should have done better.
I do agree somewhat with the argument that companies should't be forced to pay for contraception and there should be some disincentive for people who abuse their health and become a burden on society (e.g. through smoking or obesity) but a civil society should provide basic necessities of life which includes health-care. Provide is probably the problem word - it should make sure people have access to it and for some that will mean stepping in to help.
It's incredible to us, coming from the UK, that in the US people who become seriously ill can literally lose everything, their house, their life savings, trying to pay for often over-priced treatment and, when the money runs out, so does the treatment. You died. Bye.
Even regular "life" things like having a baby can be astronomically expensive if you don't have coverage. If you are found injured at the side of the road and don't have insurance? You get taken to the poor-people shitty hospital / given a big bill.
US health-care is incredibly expensive because of the broken, cherry-picked insurance system and there are huge corporates who can lobby to keep it that way. It's one of the achievements of our age that they have managed to convince so many Americans that they shouldn't want affordable and accessible healthcare for all.
Obamacare / ACA wasn't a solution because it was half-baked and relied on young, healthy people signing up to pay more to make it make sense. This is one of those things where EVERYONE has to pay into it to make it work and that is something that only government can do. It wasn't single-payer because the republicans wouldn't go along with it so that's the best they could do. Now it looks like Trump wants to just write a cheque and add it all to the deficit. Politics.
Socialized medicine is "evil" to many Americans because of the "socialized" aspect of it. But everyone is OK with socialized defence, roads and so many other aspects of life where the government legislates, taxes and coordinates things.
Political discourse in America is completely broken right now. You couldn't get people to agree what color the sky was (it would probably start with someone complaining that the word 'color' shouldn't be used because it's racist).