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.
The Gunners appealed but the FA have upheld the sending off, 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 today, 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."
The loss of Giroud is certainly a blow to Arsene Wenger's squad as they look to lockdown Champions League qualification, the £13million summer signing from Montpellier having scored 17 goals.
Arsenal retain a slender one-point lead over fourth-placed Chelsea and are two ahead of north London rivals Tottenham, albeit having played a match more.
Gunners boss Wenger will now have to rethink his attacking options for crucial games against United, then at QPR and home to Wigan before Giroud will be eligible again on the final day at Newcastle.
Theo Walcott is one option for a central striker, and is currently leading scorer with 18, while German international Lukas Podolski is another who has been pressing for a start down the middle, looking to add to a tally of 14.