Mr Cox, who is on the Reaganite wing of the party, accepted his defeat in a breakfast television interview, where he said: "The truth is that the vote is in, and Bob Livingston is going to be our next Speaker, and I am withdrawing my name for that reason."
He said he had made more than 200 calls to potential supporters, but received only 90 committed votes compared with Mr Livingston's 110. The election, which is now likely to become an acclamation, is due to be held on 18 November.
Mr Livingston, whose announcement that he would challenge Mr Gingrich effectively triggered the Speaker's resignation, has presented himself as a pragmatist capable of forging deals and furthering the Republican agenda now that their majority in the House has been cut to six.
Members of Gopac - the Republican action committee founded by Mr Gingrich to spearhead his policy efforts - spoke of their apprehension about the party's future direction.
Mr Gingrich's chief of staff, Arne Christensen, called for unity and said it was not time to "write off anyone". He also recommended that the next House leadership should work with Republicans in the Senate and Republican state governors to ensure there was a common agenda.Reuse content