A statement released by Keane on Tuesday evening said he did not think that the best interests of Irish football would be served by him rejoining the squad.
Hopes had been raised earlier in the day that Keane might apologise to Irish manager Mick McCarthy over the outburst which caused the ex-captain to be thrown out of the squad.
However, Keane's statement, sent to Irish national network RTE, made no mention of an apology.
The Manchester United star said that "the damage has been done" and that it was time to bring the saga to an end.
Keane wished the Irish management and players "all the best" in the World Cup and urged Irish supporters to get behind the team.
"I believe that the time has now arrived when I should bring to a conclusion the continuing speculation with regard to my participation in the World Cup and for the players to concentrate fully on their preparations for the competition free from all further distraction.
"I do not consider that the best interests of Irish football will be served by my returning to the World Cup.
"The damage has been done.
"I wish the team and the management all the best and they will have my full support throughout the competition.
"I urge all the people of Ireland to give their entire support to the team.
"I do not feel that any useful purpose will be served by my making any further comment.
The Football Association of Ireland's acting general secretary John Byrne said efforts to broker a peace deal had continued "up until seven o'clock tonight".
And he confirmed that an apology from Keane would have been enough to solve the dispute.
"I'd be speculating as to how close he came (to apologising) but we did feel this afternoon he could be on his way back," said Byrne.
"Unfortunately it hasn't worked out. It's unfortunate for Roy, it's unfortunate for the FAI, it's unfortunate for the country but we now have to draw a line in the sand and get on with it."
McCarthy had confirmed on Tuesday that he would consider taking Keane back - but only after an apology, and if the players agreed.
The players later indicated that they would have been happy to have Keane back on board.
"We worked strenuously to resolve the situation. People have made huge efforts to get Roy Keane back to the World Cup," added Byrne.
"Unfortunately it has come to this and I think it is now time to get on with the business of winning our game on Saturday."