The France international was dismissed by referee Andre Marriner in the final minute of Saturday's 1-0 win at Craven Cottage for jumping into a challenge with defender Stanislav Manolev.
Despite the Gunners' appeal, the FA upheld the sending off on Tuesday afternoon, so Giroud will now miss three of the final four matches of the Barclays Premier League season - starting against Manchester United on Sunday.
A statement from the FA read: "Following a Regulatory Commission hearing, Arsenal's Olivier Giroud has had his claim for wrongful dismissal rejected.
"Giroud was dismissed for serious foul play during his side's game against Fulham at Craven Cottage on Saturday 20 April 2013.
"The player's three-match suspension will therefore commence with immediate effect."