I know many, many people who were naturalized within the last few years, and not one of them has given up their original citizenship.
It's actually an odd contradiction in the law. While the U.S. would like you to give up your original citizenship, they also never, ever want any other country to be able to force a U.S. citizen to give up theirs. And so they are sort of ambiguous about it on purpose. But they recognize that they can't force you to, and in your husband's situation I think he is entitled to keep it and get his inheritance! If he has only renounced his citizenship to the U.S. authorities, that should not be valid anyway. He would have to renounce it to Indian authorities as well.
I was born in Mexico to an American mother and have dual citizenship.
To the OP: Mexico makes a distinction between being a "citizen" and being a "national." I believe that one of them carries with it the right to vote and the right to own property, whereas the other doesn't, but does allow you to live and work in Mexico without being the spouse of a citizen. It was my understanding that since my children were born outside of Mexico, I can only give them "national" status, but not "citizen" status.
But I looked into that so long ago, that it's definitely a good idea to check the consulate site, or call them.