He has denied any involvement and said it has been a "shabby and discredited witch hunt".
Mr Williamson has called for Theresa May to release details of the investigation "so everyone can make a judgement".
He has previously said he would welcome a criminal investigation as a way to clear his name, but the Metropolitan Police said the disclosure did not amount to a criminal offence.
"This whole affair hasn't been about trying to find the real culprit who leaked what was said at that meeting," Mr Williamson said, speaking to the Sunday Express.
"It has been a game of politics, it's been about settling scores and trying to prove the Prime Minister's political strength.
"The PM has spoken about compelling evidence. Well, I'd like to see it," he added.
He also accused Theresa May and Cabinet Secretary Sir Mark Sedwill of badly mishandling the inquiry and called for a probe into it.
"With the Metropolitan Police not willing to do a criminal investigation it is clear a proper, full and impartial investigation needs to be conducted on this shabby and discredited witch hunt that has been so badly mishandled by both the Prime Minister and Mark Sedwill," he said.
Assistant Commissioner Neil Basu of the Metropolitan police said he had spoken to the Cabinet Office regarding the nature of the material discussed in the meeting, but was "satisfied" that the details disclosed to the media did not "contain information that would breach the Official Secrets Act".
He said: "I have considered all the information available to me and I have taken legal advice.
"I am satisfied that the disclosure did not amount to a criminal offence, either under the Official Secrets Act or Misconduct in a Public Office. No crime has been committed and this is not a matter for the police."
According to reports in the Daily Telegraph, Mrs May overruled five ministers who expressed concern that the company's involvement might provide a route for Chinese spying and undermine allies' confidence in the security of UK communications.
Gavin Williamson was later sacked as defence secretary after the Prime Minister said there was "compelling evidence" he was behind the leak - something he denies.
In a statement, Mr Basu, head of the Met's Specialist Operations, added: "Any organisation has the right to conduct an internal investigation into conduct in the workplace. It is not a matter for the police unless a crime is alleged.
"No crime has been alleged by the owner of the material and I am clear that the leak did not cause damage to the public interest at a level at which it would be necessary to engage misconduct in a public office. It would be inappropriate to carry out a police investigation in these circumstances."
Opposition parties had called on Mrs May to refer the matter to the police for a criminal investigation, after the Metropolitan Police Commissioner Cressida Dick said Scotland Yard would not launch an inquiry unless the case is referred to them by the Government.
It was understood the information leaked from the meeting was not judged by the Cabinet Secretary Sir Mark Sedwill to be of a classification level that would require a criminal investigation.