Darwin himself didn't argue the human social aspects of his theories but many people after him have twisted those ideas for their own purposes. Someone is going to bring it up, so I might as well mention that Hitler was one of those people. However, I'll dismiss him and other such radicals because they aren't pertinent to your question.
So what we are talking about is a natural evolution of society -- the world society. Given a long enough period of time for this evolution to "do its thing", it is conceivable that in the future there is going to be far less social diversity than today. In fact, if you look at just the past 100 years or so, it is hard to miss. I call it he "MacDonaldization" or "Hollywoodization" of the planet. Differences between peoples and cultures have greatly started homogenizing into blandness. Fewer lumps in your tomato soup, so to speak (I for one like the lumps).
One caveat to what I just said is that human understanding and behavior have changed precious little in at least the last several thousand years of civilization, so in one sense our behavior seems to resit refinement. Even if one society were to exterminate all others on the planet (which Darwin theorized could happen), it would not alter basic human nature within the surviving civilization which would eventually decay, devolve and split, and do what societies always seem to do due to human devisiveness.
Getting back to your question about health care affecting social Darwinist principles, I have to argue that ultimately there is no such thing as an unnatural act. We can attach ethics and morality to an act, but if something can occur then the law of random chance says that given a long enough interval it will occur, therefore it cannot be unnatural, it can only be assigned a degree of likelyhood.
Using that as my premise and definition regarding social evolution, I have to say that instituting social healthcare in the United States cannot violate the laws of "Social Darwinism". What will be will be, and at some point a "public option" has just as much chance as any other.