The answer is that all rights - even those enumerated in the Bill of Rights - are limited. The supreme court has repeatedly stated that there is no such thing as an unlimited right.
IF there was a Right to Health Care, it would be a limited right - based on what level of Health Care could be provided given the medical resources and respected (not infringing on) the rights of medical professionals.
For example, a person who has had a heart attack may be entitled, by right, to be resuscitated, but pending the availability of medical resources, they may not have the right to triple bypass surgery.
The biggest challenge to the notion of Health Care being a right is delineating the exact limits of that right.
In some ways the medical professionals have already set de facto standards for the existence and limitations on people's rights to Health Care. I don't know if they have clearly codified these or whether it's left up to the discretion of indivdual medical professionals.
This strikes me as hopelessly subjective. In any case, I wonder if you'd advocate for all of life's necessities as 'rights' in that case? Food, shelter, clothing, transportation?
As I said, I can conceive of health care as a service that government might provide - much as it provides basic education - but it makes no sense as a 'right'. Neither does education, fwiw. To claim that a product or service is a right is to fundamentally misunderstand what political rights are all about. They are freedoms, not guarantees of service.
I agree that the right to Health care is hopelessly subjective. However that does not mean that it should not be a right simply because it's hard to define.
As far as other basic necessities, these would fall under the basic right to life mentioned in the declaration of independence. Most courts would agree that a person does have the right to steal a loaf of bread if they are starving, to break into a building if they are in fear of dying from exposure. Most municipal governments do provide food, shelter, clothing and (if it's a necessity) transportation because most people do feel that everyone should have a right to these.
These are not enumerated in the constitution because of the complexity and limits of providing these.
Everyone has a right to seek healthcare. They do not have a right to take from me to pay for another's healthcare.