It is more complicated than that. Markets have the structure and regulations we give them. Health care gets set up and delivered with very different concepts and guidance in various places regardless of whether it is for profit.
We don't have a true free-market in health care, so regulations and various programs are meant to create an environment where we get the benefits of a free market while still having universal availability and a universal "standard of care".
Of course it doesn't work very well. That's asking too much of any system.