I have two responses:
First, why 15 and not 20? Actually, why either of those and not viability (which is varies 20-28 weeks). Viability had been the cutoff previously - I'm not sure I follow the argument for moving it earlier. I know some people had expressed that this was because of how the 20 week fetus looked, but that doesn't go very far. I've seen "nervous system," but I'm not clear on why this is a better measurement of being an independent life than any of the three currently being promoted: viability, birth, conception.
Second, the vast majority of abortions past the 20 or so week mark are for rape/incest/life of mother/etc. I'm comfortable with the idea that late-term abortions should be restricted to when medically necessary (+rape/incest), but I think that "medically necessary" should be left to the judgement of the medical practitioner, and not a legislative body. A big part of the reason I think this is that most of the legislative restrictions thus far, while allegedly being for medical purposes, have had little to do with medical evaluation. In addition, this is something that would need to be made on a case-by-case basis. Legislation is entirely inappropriate for medical diagnoses and analysis.
One could create evaluation committees out of independent medical practitioners, but I very much doubt that those would either get the funding to their jobs properly (as creating a huge backlog via minimal funding would mean that nobody gets through, and "muh abortion taxes"), and even if they did, they would likely be stacked by people uninterested in objective analysis.