However, he has accused opponent Yohan Cabaye of using language which has "no place either on or off the football pitch"
The Australian's club Everton announced earlier on Tuesday that their appeal against his red card had been rejected by the Football Association.
Cahill has now issued a statement, through his management company Wasserman Media Group, in which he said: "I accept my three-match ban.
"My reaction at the end of the game was out of character but I was disgusted at comments made to me by Cabaye.
"I don't wish to elaborate on the exact detail of what was said - but his words have no place either on or off the football pitch.
"Cabaye has apologised via his club's website and as far as I am concerned the matter is now closed."
Cahill was sent off by referee Andre Marriner during his side's 3-1 win over Newcastle on Sunday and French midfielder Cabaye who issued an apology on the Magpies' official website at the weekend.
"I would like to apologise for the cross words I exchanged with Tim Cahill which resulted in his red card and my booking on the final whistle," he told nufc.co.uk.
"It was unnecessary and, regrettably, the result of frustration and disappointment, all in the heat of the moment.
"It is out of character and for that I am sorry."