Downing Street has refused to commit to publishing the full legal advice given on the Brexit deal despite a unanimous resolution by the House of Commons, in a move likely to spark a major new process row with Labour and Tory Brexiters.

No 10 has only agreed to publish a “full, reasoned position statement” – a summary of the legal advice rather than the full text – which Labour said would not comply with the terms of the Commons vote.

The shadow Brexit secretary, Keir Starmer, said the offer was completely unacceptable and Labour would use every parliamentary mechanism available to challenge it.

A cabinet source said the full legal advice on the Brexit deal presented by the attorney general, Geoffrey Cox, to ministers included an uncompromising view of the Northern Ireland customs backstop, saying that the joint UK-EU exit mechanism amounted to an effective EU veto on when the UK could leave.

A fear that this would inflame the Tory right was the reason for not publishing the full legal advice, the source added, although No 10 insisted this was not the case.

The government initially offered to publish a statement on the legal advice as a concession to MPs when it became clear Labour and Tory rebels from the Brexiter European Research Group intended to force the publication of the full advice during a Commons debate two weeks ago.

However, despite the offer from Cabinet Office minister, David Lidington, at the dispatch box, it became clear during the debate that MPs would not accept the concession.

Labour had used the arcane procedure of a “humble address”, which it used previously to force the release of Brexit impact assessments. Its demand to see the legal advice was backed by Tory Brexiters and the Democratic Unionist party, so ministers conceded defeat by ordering Conservative MPs to abstain in a Commons vote.

On Tuesday, however, the prime minister’s spokesman said a “full, reasoned position statement” is what Lidington offered to publish and that would be what was given to MPs, despite the full terms of the motion.

“The position is as set out by David Lidington in his statement to the House a couple of weeks ago,” he said. “This is for a full reasoned position statement laying out the government’s political and also legal position on the proposed withdrawal agreement and attached protocols. So the commitment remains as it was set out a couple of weeks ago.”

Asked if that complied with the Commons motion, No 10 said: “It complies with the commitment that David Lidington gave to the house.”

A source from the ERG suggested Tory Brexiters would back attempts to force publication. “Why does the government think it can ignore parliament? More to the point, how does the government think it can do this?” the source said. “No 10 is going to have to respect the will of the Commons.”

Starmer said: [Parliament] was very clear in what it was demanding – and that view was shared by MPs from all sides of the house. The government didn’t oppose what it knew to be a binding motion.”

Labour is expected to challenge the decision, which could include an urgent question or debate in the house. “If ministers continues to dig in, then Labour will use all the mechanisms available to force this information to be published,” Starmer said.

He wrote to Lidington on Tuesday asking for an explanation, saying the government must publish in full. “Labour and parliament will accept nothing short of the full legal advice presented to cabinet,” he said. “A legal summary is clearly not sufficient and will not comply with the unanimous decision made by the House of Commons.”

The House of Commons Speaker, John Bercow, has previously made it clear the government is obliged to comply with the Commons motion. He said the motion “is not just an expression of the opinion of the house; it is an expression of the will of the house that certain documents should be provided to it.”