The top Brexiteer made his comments shortly after United Kingdom opposition and Labour leader Jeremy Corbyn backed a second referendum if Labour's soft Brexit plan is voted down this week, providing a major boost to campaigners of the Final Say vote.
Labour MP for Manchester Central, Lucy Powell, claims around 25 MPs would vote against any second referendum, meaning that without significant support from Tory MPs, it would struggle to pass the Commons.
At the last election, Labour MPs stood on a manifesto of delivering the referendum result and committed to get a deal which would work best for British people.
"The UK will join the GPA as an independent member as soon as possible if we leave the European Union without an agreement, or will remain a member under European Union schedules if we have an implementation period", the statement says.
Williamson later issued a statement in which he said he was "sorry for how I chose to express myself on this issue", and that he was "trying to stress how much the party has done to tackle anti-Semitism".
The video, obtained by the Yorkshire Post, was recorded at a meeting in Sheffield in the wake of last week's resignation of eight Labour MPs to join the Independent Group.
Labour MP Julie Cooper wrote she had "no intention of voting for a second referendum".
"May's strategy shows a clear pivot from her previous strategy of "My Deal vs No Deal" toward "My Deal vs A50 Extension", and is an effort to pull hard-Brexit members of her party on board to support her deal or risk no Brexit at all (helped in part by Labour's move yesterday to support a second referendum)". May says a new vote won't be held this week and could come as late as March 12.
The European Commission, the EU's executive arm leading the Brexit negotiations, is still working on the assumption that Britain will leave the bloc on March 29, commission spokesperson Mina Andreeva told reporters earlier in Brussels.
What was Labour's Brexit stance before?
Mr Coppard, who refused to stand for election previous year after the anti-Semitism row, tweeted: "I know that the 40 members who voted to ignore the genuine concerns of the Jewish community and good people from across our movement do not speak for the majority of the members of our party in Sheffield Hallam or the community itself".
But he said the situation had evolved since then, and it would "make sense for there to be a further public vote" if the country was faced with a choice between Theresa May's Brexit deal and no deal.
He's said he supports staying in some kind of customs union and wants liberal access to (though not full membership of) the single market.
In a piece for Wednesday's Daily Mail, May said her preference was for a deal allowing Brexit on March 29 and she urged parliament to vote in favour.
Tusk said Monday that "it's clear that there is no majority in the House of Commons to approve a deal". However, these would ultimately have to be decided by parliament.
Shadow justice secretary Richard Burgon suggested there was more time for the party to push for its own version of Brexit before being forced to back another referendum. And I do not think no-deal is going to be orderly and smooth, it's going to lead to huge problems up and down the country, so it [a second referendum] needs to be seen what it's up against.