Nicholas Paget-Brown said he had to share responsibility for "perceived failings" by the authority in the aftermath of the tragedy which claimed at least 80 lives.
His deputy, Rock Feilding-Mellen, also intends to stand down from the council, which also lost its chief executive Nicholas Holgate, following a barrage of criticism over the authority's response to the fire.
Mr Khan, who welcomed the resignation, said the Government had "no option" but to appoint "untainted" commissioners who had "a genuine empathy for local people and the situation they face" to take over running the authority.
He said: "The council now needs to find a way to move forward and find a way to restore the confidence in that community.
"That can only be done with new leadership and a new approach that reaches out to residents who quite rightly feel desperately neglected.
"I ... feel the response from the council and subsequent breakdown in trust is so severe that there is now no alternative and the Government needs to step in quickly."
Commissioners were brought in to run Tower Hamlets Council in 2014, following a critical independent report into the council's award of grants and sale of properties under former mayor Lutfur Rahman.
It comes as Jeremy Corbyn said he has written to Prime Minister Theresa May urging her to increase the scope of the public inquiry.
He has asked for a two-part inquiry, the first looking at specific issues around the fire in at the 24-storey building in Kensington, west London, and reporting back soon, with an additional second part "looking at the national issues".
In his resignation statement, Mr Paget-Brown acknowledged the council had been criticised for "failing to answer all the questions that people have" but that the scale of the tragedy "was always going to mean that one borough alone would never have sufficient resources to respond to all the needs of the survivors and those made homeless, on its own".
He said: "As council leader I have to accept my share of responsibility for these perceived failings.
"In particular, my decision to accept legal advice that I should not compromise the public inquiry by having an open discussion in public yesterday, has itself become a political story.
"And it cannot be right that this should have become the focus of attention when so many are dead or still unaccounted for.
"I have therefore decided to step down as leader of the council as soon as a successor is in place.
"They will appoint a new deputy leader and cabinet."
Downing Street said the council should have "respected" a High Court ruling that the press and public should be allowed into the meeting on Thursday, the first cabinet gathering since the disaster.
Communities Secretary Sajid Javid said that it was a personal matter but "given local people had lost confidence in the leader, it is right that he has stepped aside".
Radical Housing Network, of which Grenfell Action Group is a member, also welcomed the resignation, saying it was "inexcusable that he has spent this long clinging to power".
But, it added: "His appalling resignation statement shows a dogged inability to understand the concerns of the community he is meant to represent."
Chairwoman of the Lancaster West Residents Association, Olesea Matcovschi, is also calling for it to be widened, citing concerns after members met inquiry head Sir Martin Moore-Bick and were "presented with a very narrow terms of reference".
In a separate development, Kensington and Chelsea Tenant Management Organisation (KCTMO) announced it had agreed chief executive Robert Black would "step aside" to "concentrate on assisting with the investigation and inquiry".
An interim chief executive will be appointed, it added.