I'd need to do some checking up on this to be sure, but the understanding I have is that the Geneva Convention, to which the US is a signatory, makes it illegal for us to use any weapons that continue to kill or do damage after the war has ended. It seems to me that nuclear weapons certainly must fall into that category.

Of course, that hasn't stopped us from using land mines, which also kill and do damaga after the war has ended, but if the issue is whether or not they are legal according to the international treaties we have signed, it would seem to me that they are not.