It is exceptionally unlikely you'd be turfed out if SERIOUSLY ill (ie brought in an ambulance) most western nations in the world, and especially those with public health care plans. The Hippocratic oath is what it is.

They will "bill" your insurance company if you have one and they accept it. In some cases it's too much hassle and they just don't bother. It depends on how much help you need. If it's serious and big money they can also bill your estate if you die.

For example, in Australia we had insurance and my wife saw a doctor but he said it was too much work to bill the foreign insurance companies and just to forget about it.

I had an ear infection in the UK and asked to see a doctor and the answer was the same; here's your prescription, appreciate the offer of insurance but not worth it.