An investigation found “urgent and serious problems” in the way abusive behaviour by MPs and staff is dealt with in Parliament.
Dame Laura Cox QC said a culture of “deference, subservience, acquiescence and silence” had allowed the bullying and harassment of staff in the House of Commons to thrive.
Mr Bercow has faced claims - strongly denied - that he bullied two former officials.
Sir Kevin Barron, the outgoing chairman of the Commons Committee on Standards, said the Speaker must quit.
Writing in The Times, he said: “The change in culture has to come from the top, and unfortunately I no longer believe that the Speaker, John Bercow, is the correct person to provide that leadership, so he should step down.”
David Leakey, who retired in 2017 after seven years as senior Lords official Black Rod, repeated his previous calls for Mr Bercow to consider his position.
He told BBC Radio 4’s PM programme the report showed there was a failure of leadership in the Commons.
He said: “When you add that to the individual allegations, unsubstantiated though they may be, this indicates Mr Bercow may not be fit for office and that should be taken very seriously.”
In her report, Dame Laura said it was “difficult to envisage” how the reforms needed could be delivered under the current senior House administration.
She called for the establishment of an “entirely independent process” for dealing with staff complaints against MPs in which MPs themselves play no part.
Her report painted a picture of a Commons where MPs enjoyed a “God-like status”, knowing they would never be subject to disciplinary action, and where abusive behaviour was actively covered up.
Complaints ranged from staff being shouted and sworn at and belittled on an “almost daily” basis to the “predatory” behaviour of some male MPs towards female staff.
They included frequent propositioning and “inappropriate touching” - including “trying to kiss them, grabbing their arms or bottoms or stroking their breasts or bottoms” - in an atmosphere fuelled by ready access to alcohol.
Dame Laura said while there was an “expectation of loyalty” among staff towards the institution they worked for, the standing of the House was being diminished by the failure of its senior leadership to deal with the issue.
“That sense of loyalty has been tested to breaking point by a culture, cascading from the top down, of deference, subservience, acquiescence and silence, in which bullying, harassment and sexual harassment have been able to thrive and have long been tolerated and concealed,” she said.
“This is not to demonise the entire institution, but unacceptable behaviour by some, whether elected Members or House staff, inflicts damage on everyone and undermines the legitimacy and authority of the House of Commons. Parliament is diminished.”
The calls for Mr Bercow to quit came as a Labour MP who helped block a probe into the bullying allegations against him was announced as the new chairwoman of the Commons Committee on Standards.
Kate Green was one of three MPs on the committee that refused in May to authorise an investigation into the claims.