Destro was banned by a Serie A disciplinary panel after being caught on camera lashing out at Astori during Roma's 3-0 win in Sardinia on April 6.
Retroactive bans via video evidence can take place in Italy only if the referee did not see the incident.
The initial ruling enraged Roma, who are second in the league. They claimed that the referee had seen the incident because Astori was booked for his reaction.
However, the match officials told the tribunal that they had taken no action against Destro because they had not seen the original foul.
"The appeal placed as a matter of urgency by Roma against the three-match suspension imposed on the player Mattia Destro has been dismissed," FIGC said on Thursday.
The 23-year-old will miss a fourth game after being booked later in the same match and picking up an automatic one-match suspension for accumulated yellow cards.
He will return to action on May 9 when Roma host league leaders Juventus.
Destro has scored 13 goals in 18 appearances since returning from a knee injury in December and his form had put him in the frame for a place in Italy's World Cup squad.
However, Italy coach Cesare Prandelli said this week that players guilty of bad behaviour would be denied a seat on the plane to Brazil.
"Anyone who messes around will stay at home because it will mean that they can't handle the pressure of a World Cup," he said.
"I'm sick and tired of seeing certain types of behaviour. National team players should have the strength to avoid reacting to provocation.
"In one of my first team talks, I asked whether a player who elbows another deserves to play for the national team and the players said no."
Roma are eight points behind Juventus with five matches to play. Should they beat Fiorentina they will qualify directly for the Champions League group stages.
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