There is a great deal of confusion on this point, Jonah, among pro-lifers and among people trying to understand their views. Untangling all the knots would be a major task, but let me just say a few things.

First, in answer to your question: Quite a few pro-lifers, including some pro-life scholars, believe that a sound interpretation of the Constitution would preclude legal abortion.

Second: A person can support a constitutional amendment making explicit something he already believes to be implicit in the Constitution. People who have supported a version of the Human Life Amendment that returns legislative authority over abortion to the states are not, by supporting the amendment, conceding that Roe is actually in the unamended Constitution.

Third: Quite a few people on both sides of the debate have thought–in my view, wrongly–that the due-process clauses of the Fifth and Fourteenth Amendments have something to do with the question of whether the Constitution has something to say about abortion. In the exchange to which you linked, Judge Bork makes several sound arguments against the idea that the due-process clauses forbid abortion. Unless the state is actually performing abortions, it is hard to see how it could be depriving persons of life without due process of law even granting that the word “persons” includes unborn human beings. It’s the equal protection clause that has to do the work here.

Fourth: We have been conditioned by judicial supremacy to imagine that if legal abortion violates the Constitution, it is therefore the job of the federal judiciary to end it. I am not sure how that would work in practice: Would the federal courts order state officials to initiate prosecutions of abortionists? But even if it would work, it doesn’t follow from the idea that legal abortion violates a state’s duty to provide all persons with the protection of the law that the federal judiciary must therefore rectify the situation.

My own view–which would obviously require some elaboration to defend, and is also a minority position among originalists–is that the soundest interpretation of the Fourteenth Amendment, as it relates to abortion, is that it both empowers and obligates Congress to see to it that states do not exclude the unborn from legal protection. It does not create any remedy if Congress fails to meet this obligation.