The prime minister is battling to maintain cabinet discipline as senior ministers set out rival plans for dealing with the potential rejection of her Brexit plan next month.
May has stressed that a no-deal Brexit is a possible outcome if her plan is rejected by MPs in January’s Commons showdown.
Gauke said: “I think making a conscious decision to proceed with no deal would not be the responsible course of action.”
He added he would be “very surprised if the prime minister went down that route”.
Asked if he could remain in the cabinet if that became the government’s policy, he told the BBC’s Political Thinking with Nick Robinson podcast: “I think it would be very difficult for me in those circumstances.
“I am conscious that there is a risk of an accidental no deal… Although Parliament clearly doesn’t want no deal, it’s not clear that there is a majority for a specific course of action to stop no deal.
“The best way of stopping no deal is to back the prime minister’s deal in my view.
“So I think it would be very difficult and I think if it came down to the government saying consciously, ‘well, we’ll just have to do that’, I don’t think there would be a lot of support for it.”
Gauke is rumoured to be one of a group of senior ministers who would quit the Cabinet if the UK was heading to a no-deal Brexit.
But in a sign of the increased focus on the battles to come if May’s plan is rejected, rival alternatives to the deal have been set out by cabinet ministers.
Commons Leader Andrea Leadsom said a “managed no-deal” Brexit was a possibility, while Work and Pensions Secretary Amber Rudd acknowledged there was a “plausible argument” for a second referendum.
At a press conference on Thursday, May insisted the cabinet was focused on the deal.
She said: “Everybody is very clear that not only what Government policy is but what we are all individually and collectively focused on is working to ensure that that deal is able to be agreed by and go through a meaningful vote in the House of Commons.”
May said the UK was still seeking “greater political and legal reassurances” from the European Union over elements of her plan, particularly the backstop measures designed to prevent a hard border with Ireland, before it is put before MPs in January.
But Brexiteer Leadsom suggested she had been looking at the option of a managed no-deal – with a minimalist agreement with the EU – as an alternative.
She told BBC Radio 4’s Today programme: “A managed no-deal does not necessarily mean there is no Withdrawal Agreement at all.
“This is all speculation but what I am looking at is trying to find an alternative that, in the event we cannot agree to this deal, that there could be a further deal that looks at a more minimalist approach that allows us to leave with some kind of deal and some kind of implementation period that avoids a cliff edge, that avoids uncertainty for businesses and travellers and so on.”
Gauke’s opposition to a no-deal Brexit – even a “managed” one – was clear at the most recent meeting of the Cabinet on Tuesday .
He told ministers it was “not a viable option” and “the responsibility of Cabinet ministers is not to propagate unicorns but to slay them”.
On Wednesday the Justice Secretary published a photo of himself holding a unicorn, saying it was an “unexpected Christmas gift”.