Henry's handball in the lead up William Gallas's goal handed France a place at the 2010 World Cup finals at Ireland's expense.
The debate has raged over the incident, with politicians from both countries getting involved.
The Irish FA have demanded FIFA order a replay of the game, while a spokesman from the world's governing body has suggested that will not happen.
Irish Taoiseach Brian Cowen has claimed he will gauge France President Nicolas Sarkozy's opinion, but the French Federation have insisted it was a case of luck being on their side on this occasion.
"Let's stop talking about it," said FFF chief Jean-Pierre Escalettes. "It is a mistake in our favour. The first? The last? Obviously not.
"When Given touched Anelka, the referee doesn't give the penalty. When Lloris did less than this in Serbia, he was sent off.
"There's no video. One time, it is in our favour. One time, it's against us."
The Football Association of Ireland has submitted a formal complaint to FIFA and that is currently under consideration.
However, a FIFA source has told Press Association Sport: "There is no way the game can be replayed.
"To do so would cause absolute chaos for football. If it was replayed, then every match in the future would also be subject to these calls for a replay any time a referee misses an incident.
"FIFA's rules are absolutely clear. Law five states that a referee's decision on points of fact are final. That is the end of it. You cannot replay the match on this basis.
"You have to have a rule that says the referee's judgement is always right."
FAI chief executive John Delaney had called on the governing body to uphold "the integrity of the game" after match referee Martin Hansson failed to spot Thierry Henry's handball as he fed William Gallas to score the decisive extra-time goal at the Stade de France.
Delaney's call came after the Republic party returned to Dublin still fuming at the way their World Cup dreams had been ripped apart.
Ireland led 1-0 on the night after 90 minutes courtesy of skipper Robbie Keane's 32nd-minute strike, which cancelled out Nicolas Anelka's winner in Saturday's first leg at Croke Park.
They then passed up several excellent opportunities to win the tie outright, but it eventually went to extra-time and it was then that Irish hopes were dashed.
Delaney said: "We have got to do what we have to do.
"We owe it to the players, who were magnificent. The supporters were incredible.
"It is up to the people who govern the game now. Every time I go to a FIFA congress, I hear about fair play and integrity.
"This was not a league game. This was a defining game with the whole world watching and if FIFA believe in fair play and integrity, this is their opportunity to step forward.
"From the French FA point of view, they need to look at themselves in this situation.
"Henry is their captain and a wonderful footballer, but does he want to be like Diego Maradona and his legacy to be this handball, this goal that got them to the World Cup in an unjust manner?"
Delaney added: "If we had qualified in this manner, I wouldn't be happy.
"I really believe the integrity of the game has been questioned last night."
Both Ireland's Justice Minister Dermot Ahern and assistant coach Liam Brady have called for the match to be replayed, while manager Giovanni Trapattoni question Mr Hansson's appointment for such an important game and suggested future ties should not go to extra time.
He said: "I would give the advice to FIFA - maybe in the future change the rules about the play-off. Play two games and at the end of 90 minutes, no extra time.
"I think there are mistakes in life - I have seen many situations in football. But change the rules.
"All sports lose credibility with this situation - it affects the integrity of the game.
"I will go to FIFA and advise them, 'Go straight to penalties'. It's better for football."