Obama's liability

National Review has an article detailing Obama's position during the Illinois legislature's debate over "born alive" abortions. They quote this section from Obama on the floor of the Illinois state house:
OBAMA: Yeah. Just along the same lines. Obviously, this is an issue that we’ve debated extensively both in committee an on the floor so I — you know, I don’t want to belabor it. But I did want to point out, as I understood it, during the course of the discussion in committee, one of the things that we were concerned about, or at least I expressed some concern about, was what impact this would have with respect to the relationship between the doctor and the patient and what liabilities the doctor might have in this situation. So, can you just describe for me, under this legislation, what’s going to be required for a doctor to meet the requirements you’ve set forth?
Just to be clear, the highlighted section is pretty key. Obama is objecting because of questions about the abortion doctor's "liability".

So here's the question, in a legal sense, liability refers to the right someone else has to sue you for your actions. Who would be doing the objectionable suing, and on what grounds would the doctors be getting sued?

I think it is clear that Obama is not objecting to the child suing the doctor because the doctor refused treatment and the severely-premature baby died. The baby's dead and has no advocate to hire a lawyer.

What Obama is questioning is this: will the doctor get sued by the mother for creating a live baby instead of a dead fetus? In other words, Obama's position is that regardless of the true viability of the fetus/baby and its ability to live apart from its mother, the mother has a right to a dead fetus and can sue the doctor if she doesn't get exactly that. The law, however, would require the doctor to help the infant, should it survive--thus opening the doctor up to the mother's lawsuit. So, in order protect the doctors from such litigation--to protect their right to give the mother a termination, even of a live, already-born baby--Obama essentially sanctioned infanticide.