Your interpretation of rights is off. I have a right not be imprisoned, a negative right, unless the government can manage due process. Natural rights are negative rights. I have a right not to get shot, you don't have the right to expect me to stand there and let you shoot me. Negative right require others to NOT do things like imprison me, shoot me, steal the product of my labor. Positive "rights" require others to actively do things like provide health care, housing, or candy.
Up until the 80s, health care was much more affordable. In the 70s, the whole process of having a baby cost $800, including a week-long hospital stay. The widespread adoption of employer-sponsored health insurance, largely fueled by tax regulation and the growth of Medicare, Medicaid, and welfare-based medical assistance has driven the demand entirely out of proportion to the supply, or even the need for care.
The current travesty of regulations will only make it worse, since it's now basically impossible to open or expand a private hospital.
Outrageous health care costs are a side-effect(or possibly the main goal) of government intrusion into the medical system.