I can't even figure out how diversity would factor into it. Medical treatment depends very little on people's race or culture. I don't get it at all.
That said, I do think we're moving in an encouraging direction. The percentage of the developed world that's willing to recognize others from different cultures and backgrounds as part of their own society seems to be growing, ever so slowly and fitfully. There will be years and perhaps even decades feel like regressions, but on average it feels like things are getting better.
IMO this isn't clearly true. Lots of states (especially those that currently have the worst health care outcomes) seem far more driven by the ideology that has caused their problems than by this kind of pragmatism. Example 1: the ACA's Medicare expansion.
Hell, it might even work the opposite way: proof that something worked in California could end up as an argument against it. "Candidate x wants to implement the same sort of government health care system that's taken over California. This November, send him a message that we don't do things that way here."
[0] http://www.people-press.org/2012/04/26/growing-gap-in-favora...
EDIT: also, I think this framing can badly mislead us: "If the people Mississippi don't want single-payer health care, and the people in California do, …". We get a picture in our head of Mississippi residents monolithically not wanting it while Californians monolithically do. In actuality it may be that 49% of Mississippi residents want it and 51% of Californians do. This is of course a broader topic (the benefits and failings of democracy and of various electoral systems) but I do think it's worth being aware when certain ways of phrasing might distort.
EDIT2: I don't at all mean to imply that your framing was intended to mislead; I don't think that at all.
But this is likely what will happen and we will get there eventually. If you are poor in a lower income red state you will be screwed for 20 years unless you move.
The problem with this line of thought is that the US is orders of magnitude larger, more diverse, and more complex than most European nations. For any single European nation I'd wager at least two of those are the case.
But we have a way to do this already without massive Federal overhauls - allow individual states to do what they want. This means the decisions are made closer to the people, so California can do what California wants while Mississippi can do what Mississippi wants. It also has the side benefit of California not paying for services in Mississippi.
Personally if one of the US states offered non-employer-based, single-payer healthcare I would move there in a heartbeat. If it's as successful as proposed, it would not take very long to most states to offer something similar. At that point, depending on individual implementations, it would be much easier to extend it to the Federal level.