Ms Sturgeon said: "I used today's meeting with the Prime Minister to reiterate that it can not - and must not - be a false choice between her proposed deal and a no-deal outcome, which threatens to be utterly disastrous for jobs, business and living standards".
If, against the odds, May wins, Britain will leave the European Union on March 29 on the terms she negotiated with Brussels - its biggest shift in trade and foreign policy for more than 40 years.
Pro-EU lawmakers and the DUP, which props up her government, say they will vote against, and the main opposition Labour Party says it will try to unseat her.
British Prime Minister Theresa May's government fought on Monday to defend its Brexit deal by outlining the legal basis for Parliament to support its plan to leave the European Union, but instead seemed to fan the flames of rebellion.
Ministers chose not to oppose the motion - tabled by Labour under an arcane procedure known as the humble address - as they feared a damaging Commons defeat.
The most contentious legal issue arising from the Brexit agreement is how Britain could get out of a "backstop" provision that would keep the country in a customs union with the EU to guarantee an open border between Ireland and Northern Ireland.
The 43-page document says that an Irish border backstop in the event of a no-deal Brexit would remain until a new agreement is made.
Ministers have been accused of ignoring the will of the House by only publishing a "full reasoned political statement" on the legal position concerning the Withdrawal Agreement.
The fallback plan agreed with Brussels was "not the future relationship that either the United Kingdom or the EU wants to have with one another", Olly Robbins told the Exiting the European Union Committee.
"The EU has a clear veto, even if the future negotiations stretch on for many years, or even if they break down and there is no realistic likelihood of us reaching agreement".
"If ministers stubbornly refuse to obey the order of MPs then they risk triggering a historic constitutional row that puts Parliament in direct conflict with the executive".More news: Manchester United escape in Saints thriller
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But Tory Eurosceptic Jacob Rees-Mogg told the programme: "I would say publish and avoid being in contempt of the House of Commons, which is a very serious matter".
Mr Cox, who is the Government's chief legal adviser, had staunchly defended the decision - telling MPs "there is nothing to see here". The formal advice from an ECJ advocate general - not binding but usually heeded by the court - suggested to some that revoking the "Article 50" divorce notice was an option.
"If the full legal advice is not forthcoming, we will have no alternative but to start proceedings for contempt of Parliament - and we will work with all parties to take this forward", he said. "And these are exceptional circumstances".
Commons Speaker John Bercow said there had been an "arguable case that a contempt has been committed" and ruled MPs should debate the issue on Tuesday.
In theory, the most severe penalty is expulsion from the House, although the prospects of that happening would appear remote.
Sir Keir said that if it was voted down, Labour would table a no confidence motion in an attempt to force a general election.
"Although the deal would still need to pass an implementation legislation in both houses of Parliament as well as be approved in the European Parliament (and formally also the EU Council despite the acceptance 25 November) after that, we believe this so called "meaningful vote" is the real test for Theresa May's Brexit deal".
His warning came as The Times reported that the DUP was considering abandoning the Government if there was a confidence vote, despite their "confidence and supply" agreement.
He also told lawmakers that British investment was now about 16 percent below where it was expected to be before the referendum in 2016.
"It's a false choice to say it's the PM's deal or chaos", said Conservative lawmaker Sam Gyimah, who quit as a minister on Friday over May's deal.