Speaking on Monday, Kit Malthouse said he found footage from the vigil “alarming” but added that Dame Cressida should not resign.
“Cressida Dick is an officer of superlative achievement in her life and she has been close to some incredibly successful investigations,” he told BBC Breakfast.
“I know that she is very dedicated and committed to this issue of dealing with violence against women and girls in as assertive a way as we possibly can.”
Asked later on Sky News whether he backed calls for her to resign, he said: “I don’t, and I do recognise that police are in an incredibly difficult position.”
He added that police had been in a “very difficult position” throughout the pandemic.
“We have asked them to do a job that they have never done before and to stand between the public, if you like, and this terrible virus, in a way that none of us are used to, and certainly they aren’t as well,” he said.
“So that very, very difficult position that they are in needs to be reflected in our contemplation of this. In the vast majority of cases, the police and the public have managed this situation extremely well between them.
“This particular circumstance, where emotion was running naturally extremely high, everybody incredibly distressed about what had happened, and indeed the police themselves devastated about the implications of this case.
“In these circumstances, it’s very important that we get to the detail of what’s happened and that’s what will emerge over the next couple of weeks.”
Home Secretary Priti Patel has asked the chief inspector of constabulary to conduct a “lessons learned” review into the police’s actions at the vigil, which Mr Malthouse said would assess “that everything was done in accordance with the rules”.
Anna Birley, from Reclaim These Streets, also said she did not want Dame Cressida to resign, but asked for the Met chief to meet with the group.
She told ITV’s Good Morning Britain: “We are a movement of women seeking to support and empower other women, and as one of the most senior women in British policing history, we do not want to add to the pile-on.”
Sir Ed Davey “genuinely shocked” by police scenes at Sarah Everard vigil
But she added: “We do want her to meet with us.
“We were hugely disappointed that she put out a statement yesterday without talking to any of the people who were organising the vigil and had such a difficult experience with the Metropolitan Police force.”
Labour’s shadow policing minister also said she did not back calls for Dame Cressida to step down, adding the focus “has to be on Sarah Everard and the increasing problems of violence against women”.
“I don’t think we want to be talking after this incredibly emotional week about Cressida Dick and her role,” Sarah Jones told BBC Radio 4’s Today programme.
“There’s going to be an investigation, we know that, and we need some answers, because we were given assurances and I think we all felt that the response on Saturday was the wrong one.
“But the focus has to be on Sarah Everard and the increasing problems of violence against women, the increasing problems of harassment against women, and what we can do now in this Bill today to fix some of those problems.”
Protesters march to Parliament Square and chant ‘shame on you’ outside Scotland Yard
But she added that she believed there was a lack of “sensitivity” in the policing of the event.
“I think most people will agree that police have had to make a really difficult call often through Covid and I talk to police all the time as the shadow policing minister, and I have talked about some of these challenges,” she said.
“I think there were two things that went wrong. The first was that an agreement wasn’t come to in advance with those women who really just wanted to pay their respects
“It wouldn’t have been unlawful if an agreement had been reached, that’s what the judge said, and it was unfortunate that an agreement wasn’t reached.
“Then I think it was wrong the way it was policed on the night, the sensitivity wasn’t there.
“I think it speaks now to the issue of the Government trying to rush through legislation that changes the laws again on public processions and assemblies and protests.”
Calls for Dame Cressida to resign were led by Liberal Democrat leader Sir Ed Davey, while Women’s Equality Party co-founder Catherine Mayer said her position was “untenable”.
Labour leader Sir Keir Starmer said the Commissioner should not quit, but condemned the policing on Saturday as “wrong”.
Dame Cressida said what happened to Ms Everard made her “more determined, not less” to lead the organisation, and welcomed the Home Secretary’s request for an independent investigation into the events over the weekend.
Meanwhile, the chairman of the National Police Chiefs’ Council said police require further “clarity” on how they should handle demonstrations during the Covid-19 pandemic, t
Martin Hewitt told BBC Radio 4’s Today programme on Monday that policing demonstrations and public order is always “incredibly challenging”.
He said: “You are balancing different rights, you are balancing legal regulations, you are balancing health and safety. They are very difficult decisions for commanders on the ground to make in any set of circumstances.
Jess Phillips: Metropolitan Police got it wrong at every turn over vigil response
“Where we are at the moment, where we are under Covid regulations, we have got the public health threat of the pandemic, those decisions have been made even harder for commanders.”
He added: “These are complex. We want clarity so commanders on the ground can make those decisions in what are always very challenging circumstances.”
One video posted online from the event on Saturday showed police removing women who were standing by the bandstand and onlookers could be heard shouting cries of “shame on you” and “let them go.”
The Metropolitan Police said four people were arrested for public order and coronavirus regulation breaches.
There were fresh protests on Sunday outside Scotland Yard and at Parliament Square, with demonstrators chanting “shame on you” at police.
Reclaim These Streets had organised the vigil before being forced to cancel following consultation with the Metropolitan Police, which said it would be in breach of coronavirus rules.