The striker handled the ball in the buildup to William Gallas's goal in Wednesday's 1-1 second-leg draw in Paris, earning France a 2-1 aggregate win and sending them through to next year's finals in South Africa.
"Naturally I feel embarrassed at the way we won and feel extremely sorry for the Irish who definitely deserve to be in South Africa," Henry said in a statement issued on his behalf on Friday.
"Of course, the fairest solution would be to replay the game but it is not in my control. There is little more I can do apart from admit the ball had contact with my hand leading up to our equalising goal and I feel very sorry for the Irish."
Henry's statement was issued after world football's governing body FIFA rejected an Football Association of Ireland (FAI) appeal to replay the match.
"I am not a cheat and never have been," said Henry. "It was an instinctive reaction to a ball that was coming extremely fast in a crowded penalty area.
"I have never denied the ball was controlled with my hand. I told the Irish players, the referee and the media this after the game."
BEST INTERESTS
Irish captain Robbie Keane, who scored his team's goal in Paris, responded by issuing his own statement.
"On behalf of the Republic of Ireland players, I would like to thank Thierry Henry for his statement," he said.
"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."
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.
WIN PROPERLY
Arsene Wenger, Henry's former manager at Arsenal, agreed that a replay would be the fairest solution.
"France has to say 'yes it was a handball and we offer a replay'," he said. "It is embarrassing to qualify the way we qualified. We want to beat Ireland properly and that didn't happen."
Wenger, who agreed to replay an FA Cup tie against Sheffield United in 1999 over an issue of fair play added:
"Thierry Henry is not the problem. He played here for 10 years and has always been super fair. He has come out and said 'I made a mistake, I touched the ball with my hand'.
"Football and sport in general is full of heroes who have cheated 10 times more than Thierry.
"A player of his stature has a massive pressure on his shoulders and if somebody has to
