The coach was alleged to have tried to slap a steward in the ill-tempered aftermath of the Serie A encounter, with the league confirming his "aggressive behaviour" in its disciplinary release on Tuesday.
In addition to Garcia's sanction, Roma defender Jose Holebas will serve a one-match suspension for directing a "provocative and insulting" gesture in the direction of home fans.
The club have been fined €20,000 for failing to co-operate with an investigation into the incident, while the behaviour of Genoa fans earned the hosts a €30,000 penalty.
Supporters were seen to hurl objects including bottles, coins and lighters onto the pitch in the moments following the final whistle.
Juventus coach Massimiliano Allegri has also been suspended for one match over comments he made to match officials after their 1-1 draw with Sampdoria at the weekend.
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