The deeper problem here may be though that treating an emergency patient that is dying thus requiring immediate care they can't afford could run higher costs that the appropriate preventative treatment they were unable to afford.
Morally we are obligated to save the emergency patients life but also morally we shouldn't have let their condition slip to the point it is in.
So the question isn't over whether there's a positive right to health care--- almost everyone agrees there is to some extent. The disagreement is over which kinds of care are included in the right.