The issue with health care and free markets, is that a fully free health care market would not morally sit well with a lot of people, and some sectors of health care lack the ingredients that make free markets work.
EG: in a free market, if someone has a healthcare emergency, yet cannot afford to pay for the service required, the free market would say -- dump this person in the street. Not exactly a morally pleasing choice. Free markets assume a rational and informed consumer; a person having a medical emergency is not always going to be a rational and informed consumer.
Not to say there is plenty of scope for a competitive marketplace in healthcare (there is, particularly for less critical care and wellness etc.), but certain portions really don't fall in scope in my opinion.
The problem with the US healthcare market is that it kind of pretends to be this "free market healthcare" system, while in reality being a half-baked socialized medical system built in bits and pieces over time. Some people are covered by the government (Medicare and Medicaid). The government relies on companies to provide much of the remainder (with tax incentives), but heavily regulates the type of health care that can be offered (eg HMOs). Since not everyone is employed by a large company, this leaves many gaps where people have no de facto health coverage at all. Except for emergency rooms, which (thanks to some notorious emergency street dumping instances in the 1980s) will accept anyone these days.
It's a mess. "Obamacare" merely plugged a couple of gaps. Much greater reform is needed, IMHO.
But there does seem to be a chorus of people who, with their strong belief in free markets, believe that the US healthcare system is a free market one (it's not) and consequently believe that the US healthcare system is the best in the world (anyone who has traveled knows how laughable this statement is). It's hard to get meaningful reform as long as this belief persists.