If you show up in a random hospital with an injury or disease, they are already obligated to provide care, it's just you're directly responsible for the bill.
In the US it actually doesn't work this way. Hospitals are only obligated to provide care for people whose lives are in immediate danger. Once the patient is stabilized, the hospital is free to kick them out. There are a handful of exceptions and many hospitals will provide care out of charity, but don't expect to get a half-million dollar cancer treatment unless you have health insurance or really good credit.