MPs voted 413-202 on Thursday to request permission from Brussels to negotiate an extension to the withdrawal date.
"I hope that MPs (lawmakers) of all parties will be over this weekend reflecting on the way forward", Lidington told BBC radio, adding the legal default was still that Britain would leave on March 29, unless something else is agreed.
"But, if not, we shouldn't be afraid to leave with no deal". They wanted a series of different Brexit options (soft Brexit, hard Brexit etc.) to be put before MPs in what's known as "indicative votes".
A new vote on May's deal is likely next week, when those lawmakers must decide whether to back a deal they feel does not offer a clean break from the European Union, or reject it and accept that Brexit could be watered down or even thwarted by a long delay.
European commissioner Jean-Claude Juncker said earlier this week that if the United Kingdom was leaving the EU it must do so by the European elections, on 22 May, but the draft paper states it would not be necessary until newly elected MEPs take their seats on 2 July. In previous political eras, by now she would probably have quit as prime minister or been forced out of office.
DUP MPs and Tory Brexiters are being warned by whips that rejecting May's deal a third time will lead to a big delay and a softer Brexit, nearly certainly including a permanent customs union with the EU.
Leading hundreds of people in the protest, Nigel Farage said: "The will of the people is very clear".
"Despite the increased risks facing rebel Conservative MPs, we doubt that sufficient numbers will fall back into line to pass her deal given entrenched hard line opposition", MUFG analysts wrote on Friday.
Protesters plan to set out Saturday from Sunderland, which is 270 miles (434 kilometers) north of London that voted by 61-39 percent in 2016 to leave the EU.
Prime Minister Theresa May will try to persuade MPs for a third time to back her Brexit deal over the coming days. "What it does is precisely what the word delay says, it just delays the point in which we come to that decision", remarked May at a news conference.
But he said he would stay on as a Conservative MP and the party has accepted his offer saying that he is a valued member and they hoped to continue to benefit from his drive and energy.
Support for Mrs May is at a similarly low ebb among Government ministers, with one Cabinet source telling The Telegraph that "there are only two ministers in the Cabinet who still support her".