Michael Gove should resign from the cabinet if he briefed the Sun newspaper that the Queen is a staunch Eurosceptic, Tom Watson has said.

In a move to flush out the justice secretary as the source of a Sun front-page story that the Queen supports Britain’s exit from the EU, the Labour deputy leader said Gove might be guilty of breaching the privy council’s oath of confidentiality and of involving the Queen in a politically controversial matter.

“Surely any member of the privy council who was a source of this story, or whose special adviser or ally was, stands in contempt of his privy council oath and should be removed from office if he won’t honourably resign himself,” Watson said in a Commons urgent question.

Watson piled the pressure on Gove after the justice secretary failed over the weekend to deny that he was the source of the Sun’s front-page headline last Wednesday which said Queen backs Brexit. The newspaper reported that the Queen had “let rip” at Nick Clegg during a “bust-up” on Europe at a lunch at Windsor Castle in 2011.

In a carefully crafted statement, Gove said on Saturday: “I don’t know how the Sun got all its information and I don’t think it’s really worth my adding anything to what’s already been said.” His remarks appeared to be designed to leave open the possibility that he may have been responsible for providing the Sun with some of its information about the Queen’s views on the EU.

Under the arcane rules of Commons, Watson was not able to ask a question about the Queen. John Bercow, the Commons speaker, read from the ancient Erskine May rulebook to ensure that Watson asked a narrow question to the Commons leader Chris Grayling on whether the privy council’s rules on confidentiality have been amended during the referendum campaign.

Grayling, who is also lord president of the council and one of five cabinet members who have joined Gove in supporting a UK exit from the EU, said the rules had not been changed. He added that the government saw no need for action, beyond a complaint over accuracy by Buckingham Palace to the Independent Press Standards Organisation, because Clegg had dismissed the Sun’s account of a Windsor Castle lunch as nonsense.

The leader of the Commons said: “The members of the [privy] council … swear an oath to maintain the confidentiality of these briefings. None of that has changed because of the current circumstances.”

Grayling added: “Last week a national newspaper published a story allegedly based on a conversation that took place at a lunch following a privy council meeting. However, my predecessor, the then lord president [Nick Clegg], has said very clearly that the story was categorically untrue. Buckingham Palace has referred the matter to Ipso. Given all of this, I do not believe that there is any need for further action here.”

Watson had earlier warned that Gove might be guilty of breaching the rules of the privy council after issuing a “hardly categoric” statement to distance himself from the Sun story.

Labour’s deputy leader told Grayling: “The sovereign’s constitutional impartiality is an established principle of our democracy. It is incumbent on those in political office to ensure that this remains the case. Such a breach would be particularly serious and significant.

“If the justice secretary were to have disclosed this information, he would have breached the principle of confidentiality and prayed in aid [of] the monarch in a politically controversial manner. But he would also have undermined his role as the minister responsible for upholding the rule of law. Does the minister therefore agree that the public have a right to know whether or not the justice secretary was a source of this story and please will he now urge his colleague to confirm or deny such allegations?”