Here is a picture via Marbury via 538, which is a great stats and US politics site:
It pretty clearly says SOMEthing. There's a debate on the site about lifestyle reasons for American ill-health, and so on and so forth. The basic issue this picture doesn't address, and I am by no means the only person in the world who knows this, is that it does not illustrate the massive divide between the insured and uninsured. The insured pay too much, and that is why the circle is on the right, but they almost certainly live longer as a result of their insurance than this picture implies. The uninsured contribute little to spending, have worse lifestyles and die younger, dragging the circle down the expectancy axis.
A problem when dealing with this, therefore, in the real world, is: the insured get good healthcare even if they do pay (invisibly via work on the the whole) through the nose. And I'd take a pretty strong punt that there is considerable overlap between the insured population of America and America's voting population. All of us, even if we are part of imperfect health systems, don't want to risk losing them for something that might be worse. Thus, Obama is trying to persuade insured Americans who vote to vote to pay more* to help uninsured Americans who don't vote. I think.
*In a more efficient system they need not pay more, but that is certainly one of the possibilities, and certainly in the short term, given the structure of the American healthcare system, it seems likely that this is what would happen.