The word "died" is incorrect here. I would prefer "deceased". An adjective should replace "died".

"Born in 1900 and died in 1999," is not correct. The problem here is that "(be) born" is a passive construction, and that carries over to the next verb. You are basically writing, "was died". I think it's better to rewrite the whole sentence, and make clause a proper sentence, as per Koren's suggestion. The only other option I can think of is "Living from 1900 to 1999," but it still sounds a little off.

You could say "He was born in 1900 and he died in 1999"