Some credit cards come with automatic travel insurance, but regardless it is well worth researching before you travel. Hospital stays abroad sometimes need to be paid out-of-pocket until you are reimbursed later on.
It all depends on the circumstances, your country of origin and any extra insurance you might have taken out.
I ended up in hospital for a fairly routine operation about a decade ago and had to pay out of pocket, luckily it was in a place where they just charge you for actual cost rather than to try to balance their budget by bankrupting you so it ended up being $1500. The same thing in the US would have been a large multiple.
My aunt broke her shoulder while travelling in the US and had to fly back to Ireland to get it properly treated after an initial x-ray. She would have been ruined financially otherwise.
Is that particularly different from how you'd be treated in any other foreign country?
I mean, I have no idea what would happen if I broke my arm while overseas either. I have insurance (in the US), and assume it would get sorted out somehow, eventually, but I have no idea what my share would be. I just don't anticipate being turned away at the hospital door though.
You most definitely get some huge bill. Depends what covers that bill. Travel insurance, your home healthcare or some extra insurance you buy for the US specifically.