There is no such thing. Hospitals, Doctors any health practitioner must have a Government license.
When I was young there were private hospitals, mostly charitable owned by churches. Little charity was handed out, you had money, or insurance, you got in, else you didn't. I was born at home as was my brother, doctor delivered, no idea what that cost, but my father was erring 50 cents a day at the time.
While I understand what you’re driving at, there are hospitals that are run by businesses and not by government. These would be “private”. There are also hospitals owned by religious and other organizations that are run as non-profits (whatever that means).
I assume various levels of government run clinics and hospitals, as well.