Destro was handed a three-match suspension after he was captured on camera lashing out at defender Astori during a tussle for the ball in the 33rd minute of the clash at the Stadio Sant'Elia earlier this month.
Roma emerged as 3-0 winners thanks to a hat-trick from Destro, who missed last Saturday's 3-1 success over Atalanta because of a ban that came as a result of the number of bookings the striker had accumulated.
The capital club appealed on the grounds that the referee had already made a judgement on the incident, and therefore video evidence should not have been used.
However, Roma's pleas fell on deaf ears, meaning that Destro will miss the upcoming Serie A meetings with Fiorentina, Milan and Catania.
Destro has scored 13 goals for Roma this season, and is considered to be an outside candidate for Italy coach Cesare Prandelli's FIFA World Cup squad.
But the 23-year-old will now have just two games - against Juventus and Genoa respectively - to convince Prandelli that he deserves to feature at the tournament in Brazil this June.
