Henry prevented the ball going out of play with his hand
The French Football Federation (FFF) has dismissed the Republic of Ireland's request for a replay of their controversial World Cup play-off game.
The Republic pleaded for the French to offer a rematch after Fifa ruled that the result would stand.
France striker Thierry Henry, who handled the ball in the build-up to the winning goal, had earlier said a replay would "be the fairest solution".
But the FFF released a statement ending any hope of the game being replayed.
With the decisive World Cup qualifying tie finely poised at 1-1 in the first half of extra-time, Barcelona striker Henry twice handled a long ball into the area before squaring for William Gallas to bundle home the eventual winner.
The incident in Wednesday's game attracted mass news coverage across Europe, but Henry - who admitted the handball immediately after the match - waited until Friday before releasing a statement.
In it he said: "Of course the fairest solution would be to replay the game but it is not in my control."
Both Irish skipper Robbie Keane and the Football Association of Ireland (FAI) then pleaded with the FFF to agree to replaying the game.
Keane said: "On behalf of the Republic of Ireland players, I would like to thank Thierry Henry for his statement this afternoon that in his opinion a replay would be the fairest option.
"As captain of the French team, to make such a statement took courage and honour, and all of us recognise that.
"As captain of the Republic of Ireland team, I would also be happy for a replay to happen in the interest of fair play so that whichever team qualifies, can do so with their heads held high.
"We can only hope that the French Football Federation might accept the wishes of both captains in the best interests of the game."
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It was closely followed by the FAI calling for the FFF "to join with it and the captains of both the French and Irish teams, Thierry Henry and Robbie Keane, to request a replay from Fifa that would protect the integrity of the game worldwide".
But on Friday evening the Republic's French counterparts appeared to put an end to any hope of the game being replayed.
"The FFF understands the disappointment and bitterness of the Irish players, management and supporters," read a statement. "The federation never sought to deny the refereeing error which saw the equalising French goal allowed.
"At the end of the match and because French football itself has suffered in the past by events of a similar nature, the FFF management expressed their regrets and sympathy to their Irish counterparts.
"During matches, decisions are taken by the referee and these decisions are final. As a result, the result of the match cannot be changed and the match cannot be replayed.
"The Fifa decision is binding and applies to both federations."
The controversial goal has led to former Arsenal forward Henry being labelled a "cheat" by parts of the media and suggestions by ex-players and pundits that his reputation would be tainted.
"Naturally I feel embarrassed at the way that we won and feel extremely sorry for the Irish who definitely deserve to be in South Africa," said the 32-year-old Frenchman.
"There is little more I can do apart from admit that the ball had contact with my hand leading up to our equalising goal and I feel very sorry for the Irish.
"I have said at the time and I will say again that yes I handled the ball. I am not a cheat and never have been. It was an instinctive reaction to a ball that was coming extremely fast in a crowded penalty area.
"As a footballer you do not have the luxury of the television to slow the pace of the ball down 100 times to be able to make a conscious decision. People are viewing a slow motion version of what happened and not what I or any other footballer faces in the game.
"If people look at it in full speed you will see that it was an instinctive reaction.
"It is impossible to be anything other than that. I have never denied that the ball was controlled with my hand. I told the Irish players, the referee and the media this after the game."
Henry's former manager Arsene Wenger echoed the frontman's thoughts, adding that the incident furthered the case for video technology to be used in future.
"Football accepts that a billion people see it, one guy doesn't see it, and yet it is the one who prevails. It cannot work," said the Gunners boss.
"At the game, I saw the referee giving a goal knowing that something was wrong and that is really sad.
"In the end, he gave a goal already knowing that it wasn't a goal. We cannot accept that in our sport and you have to do something about it. The referee didn't see it, I can understand that, the linesman didn't see it, but they couldn't get any help.
"For the sense of justice it is quite embarrassing to see. I think even France is embarrassed. We didn't play well at all and we won the game and won the qualification with a goal that was not a goal."
And Manchester United manager Sir Alex Ferguson also advocated the use of video technology to resolve such issues.
"The stance is that Fifa prefers human decision-making rather than technology decision-making and until they change their mind there is nothing you can do about it - you have to convince them, nobody else," said the Scot.
"It is not a matter of asking every player and manager in the world their opinion because they will all share the same one, as I do myself, that technology can play a part and can help referees in a situation like the other night."
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