"The decision by FIFA is final and should be respected by both (the French and Irish) federations," the FFF said in a statement on its website.
The FA of Ireland (FAI) appealed to FIFA for a replay after France striker Thierry Henry handled the ball setting up William Gallas's extra-time equaliser in Wednesday's 1-1 draw in Paris that sent France to the World Cup with a 2-1 aggregate win.
However, as was widely expected, FIFA wasted little time in rejecting the plea.
"FIFA has replied to the request made by the FAI to replay the World Cup play-off match between France and the Republic of Ireland," the world body said on its website.
The FFF statement said it understood the "disappointment and bitterness of the Irish players, officials and fans.
"The FFF never tried to deny that a refereeing mistake led to the French goal being awarded."
The French federation also defended Henry, saying he had spontaneously admitted he handled the ball.
Henry said in a statement released after the FIFA ruling that he felt a replay would be the fairest solution.
The FAI board met on Friday and issued a statement saying said it had called on the French federation to "join with it and the captains of the French and Irish teams... to request a replay from FIFA that would protect the integrity of the game worldwide and the pride of the French national team."
The FAI welcomed Henry's statement that a replay would be the fairest solution and has sent a reply to FIFA after the governing body rejected its appeal to play the match again.
