The FA have rejected Everton's appeal to have the Aussie's three-match ban overturned.
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."