The incident occurred in the 33rd minute of Sunday's Serie A clash at the Stadio Sant'Elia, with Destro subsequently going on to score a hat-trick in a 3-1 triumph for Roma.
Referee Davide Massa did not take action against Destro during the game, but - after the reviewing of video footage - the 23-year-old was hit with a three-game ban and an additional one-match sanction for the number of yellow cards he has accumulated following his caution three minutes before the break in Sardinia.
Roma's slim title hopes are sure to be hindered by the absence of the in-form Italy international, but head coach Garcia is confident of reversing the decision.
Speaking to LA7, he said: "The appeal? It is logical that we can win it. He is not a violent player, at all. Football is a contact sport, players fight to take position on the field and these things happen.
"The referee saw everything, he whistled against him and normally the rule states that there can be no video evidence when the referee saw the situation.
"I think now this is more than video evidence. I do not believe that Roma are being persecuted, I do not think about these things.
"The only thing that matters is doing well and if Destro is not there then there will be someone else."
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