The ICC also endorsed the decision by Hair and fellow umpire Billy Doctrove to award the match to the home team at the Oval on Sunday after Pakistan stayed in the dressing room after tea on day four in protest after being penalised for alleged ball-tampering.
"It is hugely regrettable that the match did not end with a great finish in front of a full house," ICC chief executive Malcolm Speed said in a statement on Wednesday.
He ruled out any review of the result after Pakistan captain Inzamam-ul-Haq has demanded the match be declared null and void.
"It is not the role of the ICC to overturn the decision of on-field umpires, the people who are enshrined in the laws of cricket as the sole judges of fair and unfair play, the ultimate arbiters of the game.
"In this instance, the decision made by Billy Doctrove and Darrell Hair to award the match to England was the correct one under the laws."
Speed said the ICC has received a letter from the Pakistan Cricket Board (PCB) demanding the Australian umpire should not be appointed for its future matches.
"However, it remains the role of the ICC and not our members to appoint umpires to Tests and one-day internationals," he said.
"The appointments are made without fear or favour and are based on the performances of the umpires in international matches."
Inzamam attends an ICC hearing in London on Friday on charges of ball-tampering and bringing the game into disrepute and faces a possible suspension of eight one-dayers or four Tests if found guilty.
The Pakistan skipper has warned his team could pull out of the one-day series starting next week if he is punished for ball-tampering.
Speed said: "It should be borne in mind that Friday's Code of Conduct hearing is not a political, racial or religious matter but a cricketing one.
"The hearing will deal with two separate charges, one of them the issue of the Pakistan team allegedly changing the condition of the ball and the other that it allegedly brought the game into disrepute by remaining in the dressing room when the match should have resumed after the tea interval.
"We have no vested interests in the outcome of the hearing but what we expect is that it will be fair and will illustrate our processes are suitably robust to deal with these issues," he added.
