In a statement, he said Mr Craig had "so prejudiced the interests of the party" that he should lose his membership status.
The board is meeting in Auckland on Saturday, and Mr Stringer will insist that Mr Craig's membership is on the agenda.
Mr Craig, the party's founder and main financial backer, stood down as leader on Friday and said he would "facilitate a review" of the leadership.
The Conservative Party's constitution says the leader is chosen by a board vote. Mr Craig would not be able to seek re-election if his membership was cancelled.
Mr Stringer, who has been Mr Craig's most outspoken critic, said that the board also had power to veto any nominations for the leadership.
Several members of the 10-person board, including Mr Stringer, Christine Rankin and Laurence Day, have expressed support for a change of leader.
Mr Craig did not give a reason for his resignation on Friday.
But Mrs Rankin has said one of the factors was the circumstances around the departure of Mr Craig's press secretary Rachael MacGregor two days before the September election.
Mr Stringer said former candidate Larry Baldock was sacked from the party for offences which were a "church picnic" compared to Mr Craig's misdemeanours.
Mr Baldock was suspended earlier this year after breaching the party's policy on speaking to the media.
Mr Stringer said: "I believe in even-handedness consistent with our key policy plank 'One Law For All' which Colin has championed so publicly."
Mr Craig said he is considering legal action against Mr Stringer in relation to comments made about the former leader's conduct.
No one has yet put their hand up to replace Mr Craig. Mrs Rankin and former Napier candidate Garth McVicar have ruled out running for the leadership.
Another person tipped for the job, Family First founder Bob McCoskrie, has also ruled out a bid.
Mr Stringer said he had never considered the leadership position, and there were "better and stronger options than me".
In a reference to Mr Craig's comments during the election campaign, he said: "I'm more certain that man did land on the moon and that vapour trails are not Government poison drops on the people than I am about whether I would or would not seek the leadership of our party at some later date."
Mr Stringer ran in the Ilam seat in the general election and won 1172 votes, placing fourth.