Boris Johnson said he had "boundless admiration" for his brother Jo, who has quit as a minister over Theresa May's handling of Brexit.
The former foreign secretary and his Remain-supporting brother were on opposite sides of the Brexit divide but were "united in dismay" at the "intellectually and politically indefensible" UK position.
Labour said Jo Johnson was the 18th minister to resign from Mrs May's administration since the election and it showed she was "in office, but not in power".
Jo Johnson said the choice being offered was between a "travesty of Brexit" under the deal being thrashed out by the Prime Minister and the EU, or the "untold damage" of a no-deal Brexit.
He said the public should be given the chance to vote again on whether they want Brexit, and if so whether they accept what is on offer or choose to leave without a deal.
This is not taking back control. It is a surrender of control. It does not remotely correspond to the mandate of the people in June 2016 2/2
— Boris Johnson (@BorisJohnson) November 9, 2018
Mr Johnson's sister Rachel said she was "hugely proud" of her "honourable and principled brother Jo who has put the interests of the country ahead of his political career".
Am hugely proud of my honourable and principled brother Jo who has put the interests of the country ahead of his political career https://t.co/CCXH6bbDQF
— Rachel Johnson (@RachelSJohnson) November 9, 2018
Tory former deputy prime minister Lord Heseltine, who backs the People's Vote campaign for another referendum, said: "This is a very significant resignation.
"Here is a young politician with everything to gain from staying inside the Government, pursuing his distinguished career, keeping his head down and waiting for events to take their course.
"He has not done that. He has resigned on an issue of principle putting the country before his party and his own career.
"There is a very important message in there to all Conservative members of Parliament, many of whom are appalled by what is happening but not saying what they think.
"This is the time for all of them, as Jo Johnson has done, to look deep inside their consciences, to reflect deeply on their responsibilities, and do the right thing for future generations."
Shadow Brexit minister Jenny Chapman said: "Jo Johnson is the 18th minister to resign from Theresa May's government.
"She has lost all authority and is incapable of negotiating a Brexit deal within her own party, let alone with the EU.
"Theresa May is in office, but not in power."
Liberal Democrat leader Sir Vince Cable said: "We warmly welcome Jo Johnson's support of the campaign to give the people the final say on the deal and a chance to exit from Brexit."
Former Brexit secretary David Davis, who quit over Brexit in July, said Mr Johnson was right to describe the Government's proposals as a "travesty of Brexit".
But he said a second referendum is "not the way forward and is not supported by the public".
Sorry to see @JoJohnsonUK resign. He's right that the Government's current proposals are "a travesty of Brexit" and represent a huge democratic deficit – out of Europe but run by Europe. However, a 2nd referendum is not the way forward and is not supported by the public.
— David Davis (@DavidDavisMP) November 9, 2018
SNP MP Stephen Gethins said: "Theresa May's authority is completely gone, and it is clear that her attempts to hobble together a deal can't even unite her Government, never mind the country.
"Even those at the heart of the UK government are realising what the rest of us already know.
"The only deal that can avert disaster is one that keeps our place in the single market and customs union.
"It's time for the Tories to change course before it's too late."