He was sacked as defence secretary for allegedly disclosing plans to allow Chinese telecoms giant Huawei to help build the UK’s 5G network.
Opposition MPs said there should be an investigation into whether the Official Secrets Act had been breached.
David Lidington, the PM’s de facto deputy, said the matter was closed.
Responding to questions from MPs in the Commons, he said it was not considered necessary to refer it to the police, but ministers would “co-operate fully should the police themselves consider an investigation necessary”.
Mr Lidington told MPs that Mr Williamson had not been accused of any criminal offence, but had lost the confidence of the prime minister.
Mr Williamson is quoted in several of Thursday’s newspapers as saying he was the victim of a “vendetta” and a “kangaroo court”.
He told Sky News that he “swore on his children’s lives” that he was not responsible for the leak.
Kuenssberg: Now Williamson told to ‘go away and shut up’
He also cited poor relations with Sir Mark Sedwill, the Cabinet Secretary and National Security Adviser, who carried out the leak investigation, telling the Times he would have had been “absolutely exonerated” had police probed what went on after the NSC meeting.
The inquiry into the leak began after the Daily Telegraph reported on the council’s confidential discussions – including warnings from several cabinet members about possible risks to national security over a deal with Huawei.
At a meeting with Mr Williamson on Wednesday evening, Theresa May told him she had information that suggested that he was responsible for the unauthorised disclosure.
In a letter confirming his dismissal, she said: “No other credible version of events to explain this leak has been identified.”
In response, Mr Williamson – defence secretary since 2017 – wrote he was “confident” that a “thorough and formal inquiry” would have “vindicated” his position.
“I appreciate you offering me the option to resign, but to resign would have been to accept that I, my civil servants, my military advisers or my staff were responsible: this was not the case,” he said.
BBC political editor Laura Kuenssberg said sources close to the former defence secretary had told her Mr Williamson did meet the Daily Telegraph’s deputy political editor, Steven Swinford, but “that absolutely does not prove” he leaked the story to him.
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Labour deputy leader Tom Watson told BBC Radio 4’s Today programme that Mr Williamson has “a right to clear his name” and the best way to do that is through a criminal inquiry.
He said: “This is about the law applying equally to everyone. We have had very high profile civil servants going to jail for breaching the Official Secrets Act.”
What is the Official Secrets Act 1989?
Disclosure of official information relating to security and intelligence by a “Crown servant” – including government ministers – can be illegal.
For it to be an offence the disclosure has to be damaging and done without lawful authority (ie not as part of the person’s official duties).
Being found guilty of this carries a sentence ranging from a fine to two years in prison.
If the police were to launch an investigation, the Attorney General Geoffrey Cox would decide whether there was to be a prosecution.
Former national security adviser Lord Ricketts, a crossbench peer, told the BBC’s Newsnight programme that on the face of it, the leak was a breach of the act and police ought to be considering an inquiry.
The Liberal Democrats said its deputy leader Jo Swinson has written to the Metropolitan Police Commissioner calling for a criminal investigation “to ascertain whether the actions of Mr Williamson constitute a breach”.
Scotland Yard said in a statement that it was a matter for the National Security Council and the Cabinet Office, and it was not carrying out an investigation.
“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.
Downing Street has made a very serious accusation and is sure enough to carry out this sacking.
For the prime minister’s allies, it will show that she is, despite the political turmoil, still strong enough to move some of her ministers around – to hire and fire.
Mr Williamson is strenuously still denying that the leak was anything to do with him at all.
There is nothing fond, or anything conciliatory, in either the letter from the prime minister to him, or his reply back to her.
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But former head of the civil service Gus O’Donnell told Today that the cabinet secretary had advised the prime minister that the leak was a breach of the ministerial code and not a breach of the Official Secrets Act, saying: “That is why it is a matter for the prime minister.”
Former Army chief Lord Dannatt told Today that the sacking was a “personal tragedy” for Mr Williamson, but also “very upsetting” for defence.
He said Mr Williamson had worked “very hard” for defence in trying to get more resources.
Media captionOn Tuesday, before he was sacked by Theresa May, Gavin Williamson said in a BBC interview that he had never leaked anything
The National Security Council is made up of senior cabinet ministers and its weekly meetings are chaired by the prime minister, with other ministers, officials and senior figures from the armed forces and intelligence agencies invited when needed.
It is a forum where secret intelligence can be shared by GCHQ, MI6 and MI5 with ministers, all of whom have signed the Official Secrets Act.
There has been no formal confirmation of Huawei’s role in the 5G network and No 10 said a final decision would be made at the end of spring.
The US has been campaigning hard among allies to exclude the Chinese telecoms giant Huawei from delivering 5G, because of its security concerns.
Huawei has denied there is any risk of spying or sabotage, or that it is controlled by the Chinese government.