The main reason for future in increase in health care spending is not the uninsured. It is the ageing population. In coming decades a smaller workforce has to pay for everything. Spending will increase significantly. For some info see I.O.U USA.
However there is a way to combat this. A shift in attitude is needed and it won't be easy. Here is my two cents:
People in the US need to pay more taxes. Don't kill me yet, I will explain.
Ever since the Reagan era election slogans have revolved around cutting taxes. This has made paying tax a larger mental burden than it is in most other countries. As the US used to be the land of milk and honey everybody believed that the US should have all basic social commodities usually associated with a industrialized state. The feeling however was that this could be realized without footing the bill. Now it is abundantly clear that nobody picked up the check and instead a needlessly money draining system is in place.
Looking to other industrialized countries with lower or similar GDP spending on health care and a more comprehensive health care system, the difference is obvious. These countries have higher tax rates. In these countries the notion that if you want to enjoy certain benefits like (infrastructure, health care, pensions etc.) they have to be paid for. I'm not saying people like paying tax, but it is not such red flag in front of peoples eyes.
In addition to paying more taxes a major overhaul of the system is needed. The free market principle doesn't have a humanitarian earmark. For services as health care this is sorely needed. As most would agree it is insane if in the US people would die in the streets just because they couldn't afford health care. This does not constitute a wealthy industrialized nation which upholds basic human rights. No sense in preaching freedom as a basic right when other basic rights are neglected.
Enough ranting. For a more eloquent discussion look up some of Jeffrey Sachs's views on the subject.