Milgram Experiment is one of the most interesting and controversial experiments in psychology. This experiment showed that people are willing to kill another human being who they have never met when they feel a person of a higher authority is ordering it to them, so it means people don’t feel responsible when its not them who have decided to kill that person even though it’s them who are actually doing the killing.
After the US dropped 2 nuclear bombs in Japan, the American public, were very happy and not at all upset, they even wanted to the US to drop more nukes. This is not only true for all human beings not only Americans.
Morality is a very strange subject. Most if not all people only find something immoral and wrong when it is assumed to be wrong. So it is not absolute. People could have been taught to find something totally right even though they might appear to find it extremely disgusting now. It’s depressing and a bit dark!
So to give a more direct answer to your question, American don’t give a damn because, in their community, sanctions on Iran is assumed to be moral and justified. If tomorrow the US mainstream media try to portray it as immoral and inhuman, then, the next day, you would see human right activists, protesting on street crying for the people of Iran.