Can't say for certain, but I had a US citizen relative living in Belgium. He was vacationing in Scotland. Got kidney stone attack... was taken immediately to local hospital in Scotland, then airlifted to larger hospital. Was there for ... several days, then went back to Belgium. No cost from him at any point in Scotland. Had another attack while getting back to Belgium, and ended up in a Belgian hospital for several more days.
Went to leave hospital after days of being there (and tests, and whatnot), and was told "well... you'll have to submit this to your insurance - we can't deal with it. It's going to be expensive(!)". They were sort of visibly ... agitated (possibly just because they don't deal with many overseas US people?). His total they were making him pay was around $200 USD.
I went a private hospital ER in Portugal and it worked exactly this way since I didn't have (Portuguese) health insurance. I was there a couple hours, they gave me an IV, ran a blood panel, and wrote me a prescription. I then paid the cashier €250 and that was the end of it.
Additionally, the visit actually solved the problem, which is something that rarely happens when I go to the doctor in the US.
Went to leave hospital after days of being there (and tests, and whatnot), and was told "well... you'll have to submit this to your insurance - we can't deal with it. It's going to be expensive(!)". They were sort of visibly ... agitated (possibly just because they don't deal with many overseas US people?). His total they were making him pay was around $200 USD.