Tobias Ellwood also warned the near-certain next prime minister he would be “crawling back to the table”, begging the EU for an agreement, if he crashed the UK out of the EU in October.
Mr Ellwood ducked repeated questions about his future, but attacked the claims that the UK could prosper if it tried to “run away from the EU”.
Speaking on Sky News, he ridiculed the idea that “we are able to land man on the moon 50 years ago but we can’t sort out the Northern Ireland backstop”, insisting: “We can.”
Mr Ellwood added: “If we don’t do that than the Conservative party could be destined to be in opposition for an awfully long time.
“And that is the danger that I hope every one of my colleagues will wake up to – and of course the next prime minister too.”
The attack came as Mr Hammond revealed he would walk out even before Mr Johnson reaches No 10, saying: “I intend to resign after prime minister's questions, before the prime minister goes to the Palace.”
David Gauke, the justice Secretary – another no-deal opponent – revealed he would also quit on Wednesday if Mr Johnson is the winner of the contest.
Keir Starmer said Labour would seek immediate talks with the departed ministers in the hope of building a cross-party alliance to thwart a no-deal Brexit.
Speaking to the Sophy Ridge on Sunday programme, Mr Ellwood said: “I want us to move away from discussion about no-deal at all.
“I believe a deal is absolutely possible. The fact that we keep talking about no-deal fuels the small caucus of people that see that as their destination.”
And, warning Mr Johnson would have to turn to the EU for help, he added: “You will still have to crawl back, literally moments later, to say ‘how are our financial services going to operate, how are our operations with the police going to work’.”
Sir Keir said: “On Tuesday morning some ministers will sit around the cabinet table for the last time.
“They know very well the dangers of no deal. They will have been briefed about what it would mean for jobs, the economy, our public services and the union. They will have seen the advice and read the evidence.
“After they have resigned this week, I will want to work with all those former ministers who, like me, want to ensure parliament can stop a disastrous and chaotic exit from the EU.”