I am not a lawyer, but I did spend a couple of years teaching the arguments for and against legalizing abortion. We read selections from the Supreme Court decision on Roe vs. Wade, and those selections supported @Marcus claim that the primary issue is separation of church and state. One of the primary purposes of religion is to provide simple answers to complex philosophical questions, and this issue is tied up with more philosophical questions than any other question that non-philosophers are forced to think about. Not just ethics, but Metaphysics, Philosophy of Mind, Epistemology, and Political Philosophy are all tied up with it.

If this were a question that could be answered by the scientific data, you could make a case for having the state make a decision on it. (Although even that can be problematic. Does clinical experience have the same weight as replicable Laboratory experiments? Are scientists the only people who get to decide what reality is?) When you have an issue this complex and abstract, any attempt to come up with a standardized answer will be doomed, and will end up making metaphysical commitments that are the traditional province of religions.

If we all agreed on what a person is, you couldn’t legitimately say “Against abortion? Don’t have one.” No one says “against Murder? don’t commit one.” That’s why infanticide is illegal. But there is no consensus on where personhood begins, which is why the question of whether abortion is murder becomes a matter of individual conscience. (The courts did acknowledge that the last trimester is problematic when trying to make a distinction between infants and fetuses, which is why Roe vs. Wade does permit state laws that ban 3rd trimester abortions.)

I should probably write a medium article on this, but it would probably end up being book length once I started.