Following Inter's 2-0 Italian Cup quarter-final victory at Napoli on Tuesday night, Sarri called Mancini a "poof", leading to a furious reaction from the former Manchester City boss.
Fabrizio Marrazzo, spokesman for Gay Center, has called on authorities to make an example of Sarri.
"As a Neapolitan and Napoli fan, I'm ashamed of Sarri's words," he said, adding that he wanted a meeting with Carlo Tavecchio, the Italian Federation president, and Napoli owner Aurelio De Laurentis.
"We would like football to launch a big campaign against homophobia. Such a popular sport cannot allow such violent messages."
Sarri apologised to Mancini after the match and claimed not to be a homophobe, but the Inter boss also accused him of racism.
"Maurizio Sarri is a racist and men like him have no place in football," fumed Mancini to RAI TV after the game.
"He used racist words. He started insulting me and then shouted at me, calling me a poof."
Mancini said Sarri had used the words "frocio" and "finocchio", both pejorative terms meaning homosexual.
Sarri brushed off the exchange as part and parcel of the game.
"I've seen and heard worse on a pitch," he said.
"The adrenalin of the pitch can push you to make bad jokes, but the arguments should finish at the end of the 90 minutes."
Italian media speculated on Wednesday, however, that the Napoli boss could face a suspension for his comments.
According to article 11 of the Italian Federation regulations, all "insults or discrimination... for reasons of race, colour, religion, language, sex, nationality or ethnic origin" can result in a suspension of at least four months, as well as a heavy fine.