MPs have accused one another of politicising a damning Westminster bullying inquiry, as the Commons Speaker hinted he would defy calls to quit before next summer.

On a day of high tension in Westminster, senior MPs called for John Bercow to resign immediately in the wake of the inquiry by Dame Laura Cox.

However, Labour has hinted it will not back any efforts to remove him. The shadow foreign secretary, Emily Thornberry, said it was “absolutely not the time” to be changing the Speaker and veteran MP Margaret Beckett said the party had to prioritise keeping him in post.

In her report, Cox suggested that senior house staff, including Bercow, would not command the confidence to pursue necessary changes. Bercow has denied allegations in a Newsnight investigation that he bullied his former private secretaries, Angus Sinclair and Kate Emms.

Bercow was reported to have told friends he will not resign before his intended departure date next summer, which will mark his 10th year in the post, although he has given no public confirmation.

Labour MP Kate Green, the new chair of parliament’s ethics watchdog, told the Guardian she would consider future complaints made against the Speaker but defended her decision in May to vote to block a formal investigation.

MPs on the standards committee, including Green, voted three to two not to allow a formal investigation into Bercow, after a complaint by an MP. Her unopposed election as chair has come under criticism in light of that vote.

“I do stand by my decision on that individual case, it was a difficult and finely balanced decision [as was] shown by the divided votes on the committee,” she said.

“We weighed up different factors and judged whether the substance of the complaint meant it was right for an investigation to continue. It’s really important to understand investigating an individual allegation is a different matter to what Dame Laura is talking about, which is a systemic failure.”

After the Cox report was published, both women and equalities committee chair Maria Miller and Sir Kevin Barron, the departing chairman of the standards committee, called for Bercow to go.

Speaking in the Commons during an urgent question about the report, Bercow said he backed the creation of an “entirely independent of and external to parliament” body to examine future cases.

“Independence and transparency are the best guarantors of a process which will be both be fair and command general confidence,” he said.

Miller told the Commons it was “clear that there needs to be a complete change in leadership at the most senior level, including you, Mr Speaker, as chief officer, if we are, in Dame Laura’s words, to press the reset button”.

During the debate, MPs urged parliamentary officials and ministers to take action, but many stopped short of criticising individuals. Others accused MPs of politicising the process, motivated by their dislike of Bercow’s handling of Brexit.

The Labour MP Ben Bradshaw said that during a febrile period in politics, MPs needed a Speaker who “stands up for backbenchers, stand[s] up for this house to an overmighty and overbearing executive who is prepared to drive through a Brexit that is intolerable”.

Commons leader Andrea Leadsom hit back at Bradshaw’s remarks, saying it was “not in the spirit” of the debate.

“I don’t understand why he feels the future of this great nation relies on one individual person, which is what he seems to be suggesting,” she said. Bradshaw, who is pro-remain, later tweeted that her comment “reveals her true agenda”, a reference to Leadsom’s support for Brexit.

Several Conservative MPs who have long been critics of the Speaker also spoke in the debate. James Duddridge, who previously called for a vote of no confidence in Bercow, said it was “a disturbing report of a number of unacceptable behaviours … How can we encourage Mr Speaker to stop this behaviour?”

The Labour MP Jess Phillips said she was “totally and utterly maddened”, pointing at Duddridge, whom she called “neither right nor honourable”.

“Some of us don’t actually care who is the offender, it is the victims we care about, and we will not use it for political gain,” she said.

“I think the management of this place needs a massive overhaul, the fact of the matter is that nothing I have heard today fills me with confidence that politics will be taken out of this and that the same 12 people – and we all know exactly who they are and exactly how they are getting away with it – won’t just be walking around for the next 20 years.”

Beckett later said that some Labour MPs believed they had to prioritise keeping Bercow because he had been fairer to backbenchers in crucial Brexit legislation.

“Yes, if it comes to it, the constitutional future of this country, the most difficult decision we’ve made for hundreds of years, yes, it trumps bad behaviour,” she told the BBC.

Dave Penman, the general secretary of the civil service union FDA, expressed fury at the Labour response saying the party was “happy to ignore Dame Laura Cox’s urgent calls and put party politics before people”.

The union said it was writing to the party leaders to ask that they support the recommendations made by Cox and commit to implementing them without delay.

Green said she would not allow any political calculations to get in the way of any future investigations into individuals.

“It is important complaints are taken seriously whatever the political context,” the MP said.

Bercow was reported to have told friends that he will stand down next summer, on his own terms. He had originally promised to retire after nine years in office, which would have been this summer, but he has since expressed a desire to see out the Brexit process.

A spokeswoman from the Speaker’s office said Bercow had been elected for the duration of parliament and had not made any official announcement. “In the event he has anything to say on his future plans, he will make an announcement to the house first,” she said.