Shadow Home Secretary Diane Abbott said Theresa May must face MPs following Amber Rudd's resignation.
The Prime Minister must answer questions in the Commons about her knowledge of migrant removal targets, the Labour frontbencher said.
She told BBC Radio 4's Today programme: "We want to talk to her about the aspects of the so-called hostile environment which she was responsible for and led to the Windrush fiasco."
READ MORE: Amber Rudd quits as Home Secretary as Windrush scandal and targets row proves too much
The Home Secretary quit on Sunday night after admitting she had "inadvertently" misled MPs over the existence of targets for removing illegal immigrants.
But Ms Abbott said: "I brought it to the House last week because there was this difficulty and problem about understanding her evidence to the Home Affairs select committee.
"But fundamentally the reason she had to resign was because of the Windrush fiasco.
"Somebody had to take responsibility. It happened on her watch, therefore I think it is right that she resigned."
Ms Abbott distanced herself from Labour's previous approach to immigration.
"The Labour Party is under new management now," she said.
"We are under new management. We take issues about human rights and fair rules and reasonable management of migration very seriously."
READ MORE: Amber Rudd’s resignation receives mixed reaction
Damian Green, who was sacked as First Secretary of State in December after making "misleading" statements about allegations that police found pornography on computers in his parliamentary office in 2008 which breached the ministerial code, said Ms Rudd was right to quit.
The former immigration minister said he did not know during his time in the Home Office about the destruction of the landing cards of Windrush migrants.
He told Today: "I'd never known about the landing cards at all. That's the sort of thing that happens in departments."
Mr Green said he was told about the phrase "hostile environment" in relation to illegal immigrants.
"That was just a Home Office policy which we inherited," he added.
Mr Green said Ms Rudd, who has been a key Remainer in the Cabinet, would speak "freely" now she has left government.
He said: "I'm sure that Amber will speak freely on all matters from the backbenches."
Asked if she would use her inside knowledge to inform the debate, he replied: "I'm sure she will."
READ MORE: Amber Rudd resignation signals new Cabinet woes for Theresa May
Education Secretary Damian Hinds said he hopes Ms Rudd "will be back".
He told Today: "I very much regret that Amber has taken the decision to resign but I understand why she has done so."
Ms Abbott said ultimate responsibility for the Windrush scandal rested with the Prime Minister.
"All roads lead back to Theresa May and her tenure as Home Secretary," Ms Abbott told BBC1's Breakfast.
"Many of the elements of this hostile environment originated under Theresa May and, most important of all, it was in 2014 that she passed legislation which removed the protection from deportation which up until then had applied to Commonwealth citizens."
READ MORE: Amber Rudd quits as Home Secretary as Windrush scandal and targets row proves too much
Transport secretary Chris Grayling said the string of Cabinet losses in recent months was "unwanted noise".
He told Today: "We have had some unwanted noise in the last few months, some unwanted loss of parliamentary colleagues and Cabinet colleagues.
"None of that are things we would have wanted to happen. They happen to all governments to some degree at some point or another.
"None of us want to lose parliamentary colleagues but the truth is we have got a wealth of skill in the Conservative Parliamentary party."
Asked if he was going to be made the new Home Secretary, Mr Grayling said: "I have absolutely no idea at this moment in time who the Prime Minister will appoint as the new Home Secretary."
READ MORE: Who are the contenders to be the new Home Secretary after Amber Rudd?
Mr Grayling, a Leave supporter, suggested that Ms Rudd did not need to be replaced by another Remainer.
"We are now a government that is united in wanting to deliver the best outcome for Britain in Brexit," he said.
"Okay, we have some debates and discussions about how we get there.
"But I think what's most important is she gives the right person this job because it is much more than the Brexit negotiations. It's about security and it's about the safety of our citizens."
Mr Grayling said Ms Rudd had "made a mistake" because she had not been "fully aware" of what was happening on the ground.
"She's come to the view that she should have known more, that she didn't and she's got to take responsibility for that," he said.
Mr Grayling said Ms Rudd would "never deliberately" mislead Parliament.
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