It isn't private. Before the war US healthcare was similar to most other developed countries. That is to say, it was free market, but there was some state involvement. The primary state involvement consisted of the licensing of physicians, which was (and is) done by the states and typically in accordance with American Medical Association guidelines. There were also some public hospitals in most states in an attempt to help the needy. Typically people paid for healthcare out of pocket (ie, cash) and there were insurance plans available for catastrophic events.
The real change came during World War 2. FDR had previously toyed with socialized healthcare in the 1930s but abandoned it. During the war, to control inflation, companies were not allowed to compete for workers on the basis of wages. They were, however, allowed to compete on the basis of benefits, and Congress created a new type of health insurance in which the insurer basically paid all healthcare expenses. Imagine having automobile insurance which paid for repairs and gasoline. These plans multiplied rapidly as corporations offered them to get workers. Postwar Truman again toyed with national healthcare, but decided the existing system was working. Before long nearly all Americans and their families were covered by employer provided healthcare. This new system was inherently problematic in that patients no longer had an incentive to ration their healthcare nor did doctors. Even insurers don't really have an incentive to since they can just raise premiums. Around the same time legislation was passed encouraging the construction of public hospitals, so America went from having mostly private hospitals to mostly public hospitals over the next few decades.
It must be noted now that there is some degree of universality in the U.S healthcare system, in that hospital emergency rooms cannot turn away people who seek treatment regardless of their ability to pay. This humane law, in combination with the lack of universal coverage, creates problems in that uninsured people and idiots go to emergency rooms for minor healthcare issues which severely tax critical emergency room resources and send costs soaring. This is a particularly severe problem in the Southwest due to the increasing accumulation of the latino poison there. I'm sure it is also a major public stress on states with high concentrations of out of control, feral Negroes.
The next major change to the system came with LBJ's "Great Society" in 1965. While most Americans did have healthcare, there was a problem with the elderly maintaining their insurance coverage. There was also an issue with indigents gaining access to healthcare, since healthcare was predicated on employment. Thus Medicare and Medicaid were created. Medicare is a single-payer universal insurance system, like Canada's healthcare, for all people over the age of 65. There is a monthly premium which is deducted from Social Security (America's pension system). Medicaid is a single-payer system for indigents, although each state runs its own Medicaid plan and receives matching funds from the federal government. After these changes America had close to de facto universal healthcare.
Cracks in the system began emerging soon. I am not sure why, but shortly after the creation of Medicare and Medicaid inflation of healthcare costs exploded. Right-wing intellectuals have long blamed Medicare and Medicaid, but I haven't examined their claims. There was a TIME Magazine cover article in 1979 however which did say that the system was a perfect model to inflate costs; perhaps the insurance industry had a hand in the legislation? At any rate, this problem was clearly noticed in the 1970s and since then there have been half-hearted attempts to correct it. Nixon proposed universal single-payer healthcare. In the mid-1970s Congress created a new form of health insurance known as managed care (such plans are known as HMOs), in which the insurance organization is to ration care. This did not have the desired effect but did infuriate people. As healthcare costs continued to increase, deindustrialization began, and mass immigration resumed for the first time in half a century the problem of uninsured people began to emerge. Congress attempted another remedy in the 1980s by creating legislation which allowed people to continue participating in employer-provided group health insurance plans even after no longer working there, which helped rustbelt layoffs but also contributed further to healthcare inflation and declining industrial competitiveness.
Since then the system has basically continued on autopilot. The only real changes I can think of since 1985 are the creation of tax-free medical savings plans (similar to Singapore's CPF only not mandatory). There have been two attempts to create universal healthcare, in 1993 (torpedoed by Congress) and the one Obama tried to pass a while ago. Massachusetts created universal healthcare on its own under Governor Mitt Romney, though I would consider its system to be idiotic. It should be noted that there are peculiarities of the American system which make health insurance more expensive than it should be as well. Due to separate state regulatory systems, insurance cannot be purchased across state lines (this includes all forms of insurance and not just healthcare, as far as I know).
America's lawsuit culture (and legal system generally) likely substantially increases costs, and while studies show a negligible cost increase from torts I'm not sure they take into account altered doctor behavior (e.g. performing every possible procedure to avoid getting sued, or just not seeing certain kinds of patients). While most health insurance is group (ie, an employer purchases a group plan on behalf of its employees who then copay), employers or individuals may not themselves band together to purchase group health insurance.
At any rate from this you can see that the America system is certainly not unregulated or free market, and stating that it is so is sloganeering.
I wrote this a long time ago on another (horrible, horrible) forum and just copied and pasted it here because it's kind of relevant but really because I can't stand all the bullshit rhetoric and sloganeering you hear about the U.S healthcare system, which is unusual really because ordinarily I don't really care about this sort of thing but for some reason this really irks me.