On the face of things, there is no conflict between thinking something is wrong, such as divorce and remarriage, and believing that we ought not to impose this as a restriction as a matter of law. There are many things I think are wrong that I oppose legislating against. I think you're a real slimebag if you lie to a woman in a bar in order to go to bed with her. But I don't think people who do that should be arrested.
The argument that is thrown back at people like Biden is that abortion, on the Catholic view, takes the life of an innocent human person. And, they argue, even the most minimal of governments ought to protect the lives of innocent human persons.
What Biden seems to think is that even though he as a Catholic has good reason to believe that every fetus has a right to life, he doesn't think that he has good reasons that he can provide to people who don't share his religion that every fetus has a right to life. The Jewish tradition, for example, seems for the most part opposed to the idea.
However, Catholics who disagree with Biden think that the beginning of life at conception isn't a matter of faith, but is rather a scientific fact. In other words, they not only think that abortion is in fact the taking of innocent human life and therefore unjustified, they think that good reason can be given to show people who aren't Catholic that abortion is that taking of innocent human life and therefore unjustified.
St. Thomas Aquinas, for example, argued that even though the Catholic Faith teaches that God exists, there are good reasons (five of them) that can be given to show everyone that God exists. The belief that the Universe had a temporal beginning, however, was something he thought you couldn't prove to the satisfaction of everyone, and so that was an article of faith that couldn't be proven.
Biden thinks the Catholic view of abortion is a matter of faith. His pro-life opponents think it is supportable by reason.