Labour is leading calls for detectives to investigate if a breach of the Official Secrets Act was committed when details of a highly sensitive National Security Council (NSC) discussion on Chinese telecoms giant Huawei were leaked to a newspaper.
Mr Williamson was born in Scarborough and studied at the University of Bradford.
-> Tackling climate change is like fighting a war we must not lose – Ed MilibandScotland Yard said it was not carrying out an inquiry, although would look into any information "that would suggest criminal offences have been committed".
Meanwhile Downing Street said the Prime Minister "considers the matter to be closed".
Mr Williamson's abrupt firing as head of the Ministry of Defence sent shockwaves around Westminster on Wednesday after Theresa May said the inquiry had found "compelling evidence" suggesting he was to blame.
The PM also said she was "concerned" at the manner in which he had engaged with the inquiry by Cabinet Secretary Sir Mark Sedwill.
-> Exclusive: Burberry to sell its land in LeedsBut Mr Williamson strenuously denied either he or anyone on his team was behind the leak and insisted he had fully complied with the probe.
"I volunteered everything up. I couldn't have volunteered more information on the whole thing," he told The Times.
"Frankly I'd rather have had a police inquiry, because the beauty of a police inquiry is I'd have been absolutely exonerated and would have been in the clear."
In a letter, Mr Williamson also revealed that he rejected an offer from Mrs May to resign rather than be sacked, saying that this would have sent a signal that he accepted either he or his team was guilty.
Speaking to the Daily Telegraph, the paper which first reported the leak, the former minister said he had been "completely screwed".
"She has got the wrong person and the person who did leak this is still out there," he said.
The PM's dramatic firing of her one-time close ally came on the eve of local elections across England and Northern Ireland on Thursday.
Opposition parties demanded a police probe into the affair, with Labour branding the ex-Cabinet minister "treacherous".
-> Protests by anti-fracking campaigners expected as Tour de Yorkshire 2019 gets underwayShadow defence secretary Nia Griffith told BBC2's Newsnight: "Clearly, there's a real breakdown in discipline and Theresa May now needs to take absolutely firm action, and, quite frankly, I think she needs to call in the police and have a full investigation, because we are talking, actually, breaking the Official Secrets Act and, actually, there may well be a case to answer here."
"I think it was absolutely treacherous - both to the country, but also to the Prime Minister.
"If you are going to run a government, you need loyalty from your ministers, you need loyalty from your Cabinet.
"We have here a case of yet another Tory minister putting leadership ambition before the good of the country."
Former national security adviser Lord Ricketts told the BBC: "On the face of it, this is a breach of the Official Secrets Act. All the ministers and officials around the table will have signed that.
"So, it seems to me the police ought to be considering, is there a case to be considered there?
"It would be for the Director of Public Prosecutions to make the decisions, not ministers."
Liberal Democrat deputy leader Jo Swinson has written to Metropolitan Police Commissioner Cressida Dick calling for a criminal investigation.
The letter states: "I am writing to ask you to open a criminal investigation to ascertain whether the actions of Mr Williamson constitute a breach of the Official Secrets Act."
-> Yorkshire MP demands commitment from Theresa May over future of northern transportAsked about the possibility of a prosecution, Mrs May's official spokesman told a Westminster briefing: "It is not for the Government to determine prosecutions, but the Prime Minister has said, from her point of view, that she considers the matter to be closed."
Scotland Yard said in a statement that it was a matter for the NSC and the Cabinet Office and it was not carrying out an investigation at this time.
"Clearly, if at any stage we receive any information that would suggest criminal offences have been committed, then we will look into that," it added.
The Prime Minister fired Mr Williamson in a 30-minute showdown in her Commons office which triggered a Cabinet reshuffle making Penny Mordaunt Defence Secretary, the first female to occupy the role, and prisons minister Rory Stewart replacing Ms Mordaunt as International Development Secretary.
Foreign Secretary Jeremy Hunt, commenting on Mr Williamson's departure as he arrived in Addis Ababa, Ethiopia, said: "On a personal level, I'm very sorry about what happened for Gavin's sake, but given the gravity of the situation, there was no other alternative outcome."
South Staffordshire MP Mr Williamson, 42, was a surprise appointment as defence secretary in November 2017 after a meteoric rise which saw him enter the Cabinet without ever having served in a junior ministerial role.
He was one of Mrs May's closest allies after she made him chief whip on entering Downing Street in 2016.
But during his time in the Cabinet, he showed increasing signs of independence from the PM and was widely regarded as preparing for a tilt at the top job when she stands down.