Sir Alan Duncan has quit as a Foreign Office minister in protest against a possible Boris Johnson victory in the Conservative leadership race.
In his resignation letter, Sir Alan described Brexit as "a dark cloud".
He told the BBC he quit to demand an emergency Commons debate to give MPs a chance to say whether they supported Mr Johnson's "wish to form a government".
The request for a debate - which would not constitute a binding no-confidence vote - was rejected by the Speaker.
Mr Johnson is the frontrunner in the contest which has seen him go head-to-head with Jeremy Hunt for Tory Party members' votes.
The ballot closes at 17:00 BST - the winner will be revealed on Tuesday morning and will become prime minister on Wednesday.
Sir Alan told the BBC's Laura Kuenssberg he did not have any personal animosity towards Mr Johnson and "wanted him to succeed".
But he said he was worried by the ex-foreign secretary's "fly by the seats of his pants, haphazard" style and feared Mr Johnson was going to go "smack into a crisis of government".
Mr Johnson's ability to command the support of a majority of MPs was "untested and in doubt", his former colleague said.
By establishing this one way or another on Tuesday - after the leadership contest result was announced but before the winner took office - Sir Alan said it would prevent "complete constitutional mayhem" at a later date.
"If he (Mr Johnson) has got the numbers to govern, then he can and should govern. But if he has not, in our constitution he cannot."
Sir Alan said he was "bewildered" by the Speaker's decision to refuse his request.
3. Sir Alan astonished that Bercow has turned his request down, and fears much bigger meltdown after recess when next PM can't hold a majority in the Commons - his critics may reckon he was trying to kill off a Johnson govt before he's even gone to the Palace— Laura Kuenssberg (@bbclaurak) July 22, 2019
A spokeswoman for Mr Bercow said requests for emergency debates were "strictly private" and the Speaker's Office never confirmed nor denied them.
Sir Alan's resignation came after Chancellor Philip Hammond and Justice Secretary David Gauke said they also intended to quit if Mr Johnson was elected Tory leader.
In his resignation letter to Theresa May, Sir Alan said it was "tragic" her government had been dominated by "the dark cloud of Brexit" - which he said had stopped the UK becoming the "dominant intellectual and political force" in the world.
He praised Mrs May for her "faultless dignity and an unstinting sense of duty", adding that she "deserved better" than to have her time in office "brought to an end" in such circumstances.
I resigned as Foreign Office minister this morning. Here is my letter to the Prime Minister. pic.twitter.com/1kpt7rHsF0— Sir Alan Duncan MP (@AlanDuncanMP) July 22, 2019
Sir Alan also discussed his own record at the Foreign Office in the letter, and said he remained "deeply upset that some fruitful discussions I had initiated about the possible release of Nazanin Ratcliffe were brought to such an abrupt halt".
As Foreign Secretary, Mr Johnson was criticised for his handling of the case of Ms Zaghari-Ratcliffe, a British-Iranian woman serving a five-year sentence in Iran for alleged spying.
Theresa May thanked Sir Alan for "the support you have shown me, not just during the last three years, but over the many years we have known each other", and praised his "devoted and energetic service".
Sir Alan has long been a vocal critic of Mr Johnson, once describing himself as his "pooper scooper" at the Foreign Office, clearing up mess he had created.
Most recently, Sir Alan attacked his former boss over the resignation of Sir Kim Darroch, the British ambassador to the US, who stepped down after comments criticising President Trump's administration were leaked.
Sir Alan said Mr Johnson - by failing to give his support to the ambassador - had "basically thrown our top diplomat under the bus".
He has also previously said Mr Johnson was "the last person on Earth who would make any progress in negotiating with the EU at the moment".
And in 2018, he described an article - in which Mr Johnson said Theresa May had "wrapped a suicide vest" around the British constitution - as "one of the most disgusting moments in modern British politics".
The BBC's Norman Smith said that in the resignation of Sir Alan - and the promises to quit by Mr Hammond and Mr Gauke - we were beginning to see the basis of a Tory opposition to Mr Johnson on the backbenches.
He said they - and potentially others to come - felt they could not support a prime minister comfortable with no deal and so it was better to walk now than be pushed later.
In an interview with Conservative Home, Mr Johnson said every member of his cabinet would have to be "reconciled" with the policy of leaving on 31 October - with or without a deal.
Mr Hunt has said he too is prepared to leave with no deal, but would accept a further delay, if required, to get a new withdrawal deal.
Sir Alan's resignation was criticised by Tory MP and ex-minister Greg Hands, who tweeted: "In my view, pre-emptive ministerial resignations (If reports are true) in case your own democratically-elected party leader is not to your liking are absurd.
"And I say that as a committed Jeremy Hunt supporter. Such moves make a Corbyn government one step more likely."
Who is Sir Alan Duncan?
He became MP for Rutland and Melton in 1992 and served as a shadow minister between 1998 and 2010.
When the coalition government came to power, he was appointed international development minister - a position he served in until 2014.
In 2016, Theresa May made him a Foreign Office minister - where he served under Boris Johnson.