The Commons leader has also implied that the Speaker may be sexist on Twitter.
"When I became an MP three years ago I was determined that I would not become part of the establishment".
Mr Bercow came under fire for calling Mrs Leadsom a "stupid woman" in the Commons in May past year.
"These are political assurances but there's nothing more we can do".
In an effort to assuage MPs concerns, May has lobbied her European counterparts and officials in Brussels to make concessions on the clause.
"How "livid" John Bercow's unprecedented Brexit intervention paves way for constitutional crisis", claimed The Daily Telegraph.
Simon Usherwood, a reader in politics at the University of Surrey and deputy director of the United Kingdom in a Changing Europe group, told Al Jazeera that Wednesday's events demonstrated the "degree to which the government is not in control of the Commons". History and precedent did not allow for amendments to such a motion, was the government's firm view.
"The prime minister will be updating parliament tomorrow and she will be talking about the clarifications, the reassurances that parliament is seeking that the backstop will not be permanent".
However, there was little sign Mrs May's charm offensive with Tory MPs had yet to secure further backing for her deal, as the prime minister hosted a second drinks reception in Number 10 on Wednesday night.
"Bercow was told twice by the clerks that the amendment should not be accepted".
In a statement outside the Commons, he said the vote in support of the amendment was "an important step to prevent a no-deal Brexit". "He will end up damaging democracy".
"So if her deal is defeated next week, as I hope and expect it will, will the Prime Minister do the right thing and let the people have a real say and call a general election?" he added.
Outlining his decision to select the amendment, Mr Bercow said: "My understanding is the motion is amendable, I'm clear in my mind about that".
The No 10 source said the Prime Minister had always meant to "respond quickly" if she fails to secure the support of the Commons.
"We are surprised the government amendment was selected; the advice we received was that it would not be in order", he said. "I can not allow debate but I have selected the amendment". However, pro-European Tories and opposition MPs backed the Speaker's decision.
"I would suggest to some of my honourable friends - the ones that are getting somewhat overexcited - that perhaps they should don a yellow jacket and go outside", he said.
He added: "For many of us we will now have an unshakeable conviction that the referee of our affairs, not least because you gave your opinion and your vote on the issue of Brexit publicly, that we will have an unshakeable conviction that the referee is no longer neutral".
While 70% of MPs from all parties were critical of the May government's handling of the Brexit negotiations, none of the obvious alternatives to her deal agreed with European Union leaders in November were palatable to them either.
With less than three months before Britain is due to quit the European Union, parliament began a five-day battle over May's Brexit plan with a show of force - undermining her preferred timetable if lawmakers vote down her blueprint next Tuesday.
However, she refused to comment on whether she has lost confidence in the Speaker.
But facing more than an hour of points of order in the Commons on Wednesday, Bercow stood by his decision.
Mrs Leadsom said Mr Bercow's future was a matter for him but asked if she was questioning his impartiality, she said she had the "greatest regard" for the office of Speaker.