The now former Home Office official issued a damning criticism of Ms Patel on Saturday, accusing her of mounting a "vicious and orchestrated" campaign against him.
In his televised statement yesterday, Sir Philip said he’d received reports of the Home Secretary "swearing, belittling people and making unreasonable and repeated demands" towards staff.
Labour leadership candidate Sir Keir Starmer called on Ms Patel to explain the explosive allegations before MPs.
He said: “The Home Secretary has a duty to come to Parliament on Monday to explain the allegations made about her own conduct.”
Sir Keir also called for Cabinet Secretary Mark Sedwill to start “an immediate investigation” into the circumstances surrounding Sir Philip’s departure.
“There are now urgent questions that must be answered and steps that need to be taken,” he added.
Meanwhile, health secretary Matt Hancock defended his Cabinet colleague, telling Sky’s Sophy Ridge on Sunday programme: "Priti is a very determined person but she's also very courteous.”
"Priti is a very determined person but she's also very courteous," @MattHancock says of the home secretary, who is facing bullying accusations and of ‘orchestrating a vicious campaign’ against’a top civil servant.
Follow #Ridge live: https://t.co/bFXOdYFYmN @SophyRidgeSky pic.twitter.com/fpNJY9ZBnu— Sophy Ridge on Sunday (@RidgeOnSunday)March 1, 2020
Sir Philip’s resignation also led to calls from opposition MPs and the senior public servants’ trade union for the Prime Minister to put a stop to what they claim is a campaign by his chief adviser Dominic Cummings to undermine the civil service.
It comes two weeks after Sajid Javid quit as Chancellor when the PM ordered him to fire his team of aides.
In his statement , Sir Philip said the campaign against him included “false” claims that he had briefed the media against the Home Secretary.
He said: “The Home Secretary categorically denied any involvement in this campaign to the Cabinet Office.
“I regret I do not believe her. She has not made the effort I would expect to dissociate herself from the comments.
“Even despite this campaign I was willing to effect a reconciliation with the Home Secretary.
“But despite my efforts to engage with her, Priti Patel has made no effort to engage with me to discuss this.
“I believe these events give me very strong grounds to claim constructive, unfair dismissal and I will be pursuing that claim in the courts."
Sir Philip said his experience “has been extreme but I consider there is evidence it was part of a wider pattern of behaviour."
“One of my duties as Permanent Secretary was to protect the health, safety and well-being of our 35,000 people," he said.
“This created tension with the Home Secretary, and I have encouraged her to change her behaviours.
“I have received allegations that her conduct has included shouting and swearing, belittling people, making unreasonable and repeated demands – behaviour that created fear and that needed some bravery to call out.
“I know that resigning in this way will have serious implications for me personally – the Cabinet Office offered me a financial settlement that would have avoided this outcome.
“I am aware that there will continue to be briefing against me now I have made this decision, but I am hopeful that at least it may not now be directed towards my colleagues or the department.
“This has been a very difficult decision but I hope that my stand may help in maintaining the quality of Government in our country – which includes hundreds of thousands of civil servants, loyally dedicated to delivering this Government’s agenda.”
Cabinet Secretary Sir Mark issued a brief statement about two hours after Sir Philip quit saying he had accepted his resignation “with great regret” and adding: “I thank him for his long and dedicated career of public service.
“Shona Dunn, the Second Permanent Secretary at the Home Office responsible for borders, immigration and citizenship, will become Acting Permanent Secretary with immediate effect,” Sir Mark said.
“The Home Office’s vital work to keep our citizens safe and our country secure continues uninterrupted.”
Shadow chancellor John McDonnell, asked if Ms Patel could stay in her post, told Sky News’ Sophy Ridge on Sunday: “I can’t see it, it’s bizarre.”
He added: “One of the ways the Prime Minister could go forward is to have an independent investigation but during that period you’d have to suspend the Home Secretary whilst that went on."