A couple of factors, mainly a fear of tax and years of Republican rule.
Having worked in the public health systems of both the United Kingdom and Australia I know the limitations of 'free' heath care. It is far from perfect but it works and efficiently uses resources.
It is the only civilized way to care for the people of a country. On the international stage the health care issue ranks a podium finish on great American embarrassments.
You don't have to be a communist to want your fellow Americans to be looked after, regardless of income.
The current system has handed too much power to private health providers and artificially inflated prices.
In answer to your question - the US can do it if they have the will and the poiticians think about what is good for the peolple rather than continuing with petty squabbling. I have my fingers cross that Obama will stand up to the health insurers.