The only way for the Conservatives to "regain any credibiltiy" is by delivering Brexit on October 31, according to Peter Bone.
The Tory MP told talkRADIO's Julia Hartley-Brewer that the latest deadline should be respected.
"The reason you have a date is to make sure everyone works to it," he said.
"The problem the Conservative Party has was that it did not deliver Brexit on March 29, therefore it has to deliver Brexit as soon as possible to regain any credibility."
On the leadership contest, Mr Bone said he was "happy to serve under whoever the party elects".
However, leadership contender Michael Gove said yesterday that he "was not wedded" to the October 31 Brexit deadline.
The environment secretary told an event in London that the UK should not be held to a fixed date, and would be happy to extend it by a matter of weeks in order to secure a good deal
Mr Bone continued: "If the Conservative Party didn't deliver Brexit by then, goodness knows what would happen to us in the future.
"It seems to me that as the EU have decided that October 31 is the date, then that is what everyone has to work to and deliver Brexit by that date."
The October 31 deadline was set after Brexit day was pushed back to April 12, following the initial March 29 exit date.