Mancini has failed to return Italy to the pinnacle of world football after they were beaten 1-0 at home by North Macedonia on Thursday.
The 57-year-old took over the Azzurris following their failure to qualify for the 2018 FIFA World Cup in Russia.
Mancini led Italy to qualify for the 2020 European Championships, winning all 10 group stage games before going on to hoist the tournament itself by beating England on penalties last July.
But just eight months on, Italy are back to square one after missing out on the world cup for a second consecutive time.
Italy were consigned to the play-offs after finishing behind Switzerland during the qualifiers.
Mancini signed a new contract with Italy until 2026 in May last year and Gravina expects the former Manchester City and Inter Milan manager to remain in charge for the foreseeable future.
"I hope Mancini will continue with us. He has a commitment to the project," Gravina told Rai Sport.
"I hope, like all the other Italians, that he will dispose of the waste from this elimination and remain at the helm to continue our work. I still have a lot of energy.
"I'm sorry, I'm truly embittered for all our fans. The great joy of the previous summer remains, but also the great bitterness of this defeat. We didn't expect it.
"The guys were splendid and will continue to be so. They gave our country an extraordinary dream just a few months ago. However, tonight's defeat makes us understand there is something we need to do."
