May's proposals for a future European Union relationship after Britain departs from the bloc next March had taken two-years of internal government wrangling to agree, but within 48 hours Johnson and Davis had resigned saying they could not back the plans.
"I'd have to do something I didn't believe in", he told the BBC.
Ms Caulfield supported Leave in the European Union referendum, and Mr Bradley backed Remain.
He has served as health secretary since 2012.
"The prime minister knows it's a proposal and not an end agreement".
The resignations came just days after May announced Saturday that she had finally united her quarrelsome government behind a plan for a divorce deal with the EU.
They are ones that want to see a "hard Brexit"-Britain crashing out of the EU's single market so long as it can control its borders".
Like Mr Gove - who infamously made a rival bid for the leadership after the referendum - Mr Johnson has always been suspected of aiming to lead the party and be prime minister.
British Prime Minister Theresa May moved to reassert her authority on Tuesday after two top cabinet members quit and launched broadsides against her Brexit plans, winning support from many of her ministers including a leading eurosceptic.
If Davis' resignation rattled May, Johnson's shook the foundations of her government.
'As EU negotiator I will negotiate only with the British government... so our next negotiations will be Next Monday with the British delegation appointed by Mrs May'. Gauke said. "The challenge is all very well for people to say I wouldn't do this".
The "common rulebook" plan "hands control of large swathes of our economy to the European Union and is certainly not returning control of our laws in any real sense", he wrote to Mrs May. "Others don't have that same responsibility". Under Conservative Party rules, a leadership race could be triggered if at least 48 of her MPs declared their support for one.
"Friday's announcement was turning red lines into a white flag, and David Davis has made that so clear in his resignation letter", Rees-Mogg said.