The right to abortion is based on a right to privacy. If, however, there is a single-payer health system in the United States, then the government will control all the medical access of its citizens. The government will decide what healthcare you can and cannot have. Since they will be paying, the government will have all your medical records and your complete medical history. Now further supposing this to be a mandatory system with no opt-out, people will no longer have much in the way of privacy regarding their own bodies.

President Obama has said publicly that he favors a single-payer system, and that certainly seems to be the endgame of healthcare reform. But if people do have a right to privacy, how is it possible that a single-payer health system could pass constitutional scrutiny?