In nature, there are no rights. So the discussion is not really about rights. There is no right to private property, but people band together, among other things, to create that “right”. There is no right to police, nor for the protection of an army. Not so many hundreds of years ago, if you wanted police protection, you paid a private person (or organization) for that protection. In England, Sir Robert Peel created the first general police forces (that why cops in England are called “bobbies”, after Sir Robert). During the mid 19th century, it was a burning issue in city after city in the US over whether the city should provide water and sewage. Previous to that, your sewage went onto the street and water was your own responsibility. When civic water and sewage were implemented and real-estate taxes instituted to pay for them, property owners screamed–wait for it–socialism. Of course, it is, so are police and armies. It is just a question of which services are socialized. In that sense, we are all socialists, even libertarians, or at least the one I knew once who argued that the only government services that should be provided were police and military. It is only a pragmatic question of what to socialize.
The point I want to make is that asking of this is a right is the wrong question. The right question is should it be made into a right, just like police protection, military protection, public schooling, roads, water, sewage,… To my way of thinking the justification of police to protect our bodies is no more basic than the provision for health care. Besides, the experience of Canada (where I have lived for more than half my 71 years) and many European countries demonstrates beyond a doubt that it is considerably cheaper to socialize medicine than to leave it private.
When I was growing up in the US, most hospitals were non-profit, often either religiously affiliated or publicly run. As I open the NYTimes Sunday magazine this morning, I see that page 15 is a full-page ad from a hospital, advertising for patients! Can this be? Obviously this hospital, which from its name I infer was once religiously affiliated, is making enough profit that it can afford full-page ads. I have seen many such ads recently, including in national magazines, though the clientele for a hospital can be only local. However, all this raised only pragmatic objections.
So my response to the OP is no, it is not a right in the US, but it ought to be. And it is in every other developed country and a lot of underdeveloped ones too.