Not relevant. Health care is a commodity like anything else. Is someone else's misfortune (ie being sick) your moral responsibility? I don't think so. For Example, if someone else engages in behavior that leads to AIDS, then should you and I be obligated to pay for treatment? If someone sends their money back to Mexico and then applies for Medicade should they get it? If people download children that they can't afford why should we pay for their education? This is socialism - the redistributionist philosophy that bankrupted every notion that has persisted in following it.
As far as the actual debatable point goes, it is easy ideologically to take your position. It is more difficult practically, because there are some situations that I just don't think our populace is going to accept. You are correct that some people are responsible for their own health care crises. But some are not. I don't think that we, as a nation, are prepared to say "tough sh#@" to a child brought into the ER by a parent without the resources to pay for health care. We may be, but I don't think so. So, at present, those who show up to a publicly funded ER get care, whether they can pay or not. If this is our standard, I'd much prefer that we provide efficient care to those persons, rather than crisis care. It is cheaper and more responsible to do so. For me, the only way around this conclusion is to decide that we are going to turn those folks away at the ER.
Should we segregate such care between those who are simply unable to provide for their own health care needs (children, disabled, elderly) and those who can? Probably. But I'll be amazed if we ever get to the point, as a nation, where we decide that those who can't pay for health care can just go ahead and die.