To me it would seem that such a system would have to be declared unconstitutional in the US—deprivation of life and liberty without due process of law.
In addition to that, if you can’t regulate abortion because the right of privacy includes a right to obtain an abortion, then surely you can’t regulate the right to basic healthcare, or even extraordinary healthcare like cancer care, heart transplants, etc. How can the Supremes possibly uphold a system which does that while telling government that it has no right to interfere in a woman’s decision to get an abortion?
George Orwell called it doublethink. I call it good old fashioned hypocrisy.