The Azzurri succumbed to a sensational 1-0 defeat to North Macedonia in a play-off semi-final in Palermo on Friday, ending its hopes of securing a place in the tournament in Qatar this year.
Having missed out on a spot at the 2018 World Cup in Russia, Italy will not feature in two successive editions of the competition for the first time in their history.
However, with Mancini having led the team on a 37-game unbeaten run that included Euro 2020 glory and only ended last October, Gravina still believes he is the right man for the job.
"I hope Mancini will continue with us. He has a commitment to the project," Gravina said.
"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."
Gravina wanted the previous round of Serie A fixtures be postponed to give Italy more time to prepare for the game but his request was denied by the league.
"I'm sorry the boys only met for one day to prepare for the game. It doesn't help, but I don't want to cause a controversy," he added.