Musgrove, a Democrat, served one term as governor, from January 2000 to January 2004.
He first reported his change of heart in an essay this week on the Huffington Post. He also said same-sex couples should have the right to marry.
Musgrove, who's Baptist, told The Associated Press on Friday that his evolution occurred over time. He said reading the U.S. Constitution made it clear that the government cannot deny rights to groups of people.
He commended Republican U.S. Sen. Rob Portman of Ohio, who recently announced support for same-sex couple after learning his son is gay.
Mississippi voters in 2004 adopted a constitutional amendment banning gay marriage.