The Constitution does not grant or define your rights. That is a common misconception. The Constitution primarily restricts the powers of government. Your rights are based in the principle of self ownership. Only totalitarian nations have constitutions that grant and define (thus limit) rights.
Now obviously though self ownership gives you the right to say what you want, be who you want, do with your body what you want, enjoy the fruits of your own labor and so on, one can't argue that self ownership entitles them to health care. So health care if it is a right is not a right that stems from self ownership.
So if health care is a right, it is a right based in the moral notion of Human Rights or Humanitarian Law. Humanitarian law is much different than self ownership in that humanitarian law can actually compel action by an individual. For example, if you witness an accident you are compelled by humanitarian law to assist a victim that needs your assistance. Let's say a child is seriously injured and comes to your door asking for help. If you do nothing and allow them to simply bleed to death on your lawn without doing anything or contacting emergency services, you can be prosecuted simply because you did nothing. In such a case, that child would have a right to assistance by you even though that right obviously is not derived from self ownership. Its a human right, based in humanitarian law rather than self ownership and just like many other rights we enjoy, its found no where in the constitution because as I pointed out, the constitution does not grant rights, it primarily restricts the powers of government.
So the real question is whether or not health care is a human right. Emergency care is almost universally held to be a human right. A hospital cannot deny you treatment for life saving care because life saving care is considered to be a human right even if you cannot afford it.
Now some have pointed out that a need is not necessarily a right and used housing and food as an example. The problem with that argument is that its not as black and white as they would like it to be. For example, if we had tens of millions of Americans that could not afford housing or food, we would probably be reevaluating as a society how we provide housing and food. The reason for this is that there it is implicit in the social contract that underlies all free societies that basic human needs have to be accessible to the majority of the individuals in that society. The individual does not have the right to be given free food in a society where the vast majority of people can afford food for themselves. However, if the availability and cost of food were such that a significant percentage of people could not afford it, then the general welfare of the people kicks in and the society due to human rights is compelled to find some sort of a solution whether its public or private.
So the point in all of this is that whether or not health care is a right is not simple enough to be summed up in a bumper sticker slogan.