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.
Downing Street said it had "compelling evidence" and the matter was closed.
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.
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.