Mr Rees-Mogg insisted he fully supports Theresa May, saying there needed to be a period of consistent leadership as the country leaves the European Union.
His comments to LBC Radio come just a week after he was grilled on ITV's Good Morning Britain for his views on abortion and gay marriage.
He was widely condemned by the media for sticking to traditional Church teaching on the issue.
The North East Somerset MP admitted so-called Moggmentum was "great fun while it lasts" but he had no wish to become leader.
He said: "I have no wish to become leader of the Conservative Party. I'm fully supporting Mrs May.
"I think it's extremely important that we have the same leadership for this period in our country's history, to make sure we get through Brexit properly and then go beyond that so that we can get the benefits of Brexit, which will be freer trade, lower prices in the shops, control of our own borders and of our own regulations, and we need stability to get that.
"I am completely backing Mrs May and no one serious thinks that I am a credible candidate."
Asked what Cabinet job he would most like, Mr Rees-Mogg said: "That's not going to happen."
Mrs May was asked in a radio interview whether she might promote him to the Cabinet, and "she laughed for the longest amount of time she has laughed since the general election", Mr Rees-Mogg told LBC.
"I was delighted to bring some happiness and joy to our distinguished Prime Minister, but that's how seriously she takes it," he said.
On recent controversies he said it was "a choice for parents" if they sent their son to school in a dress, described easier access to the morning-after pill as "a great sadness, because life begins at the point of conception" and said he would be happy to eat chlorine-washed chicken from the US.
Challenged over the growth in food banks under the Conservative Government, Mr Rees-Mogg said this had happened because of a change in rules allowing JobCentre staff to inform clients of their availability.
"To have charitable support given by people voluntarily to support their fellow-citizens I think is rather uplifting and shows what a good, compassionate country we are," he said.
"Inevitably, the state can't do everything, so I think that there is good within food banks.
"The real reason for the rise in numbers is that people know that they are there and Labour deliberately didn't tell them."