No one ever said that... and apparently you've entirely missed the ProChoice argument. Does a government have the right to prevent you from having an appendectomy? Where does the government end and you begin? ProChoice is about limiting absolute government authority, like, over your cells and junk.
btw the rights in question are already in the Constitution. We generally refer to it as the Fourth Amendment, though an explicit line limiting what the government can do also lives in the Third Amendment. With the 3rd and the 4th Amendments, any ban on abortion is necessarily unconstitutional, in that we have a right to be secure in our persons free from government intrusion, and the government can't force anyone to take on a tenant, even if for only 9 months.