Lawmakers rejected the deal 391-242, ignoring May's entreaties to back the agreement and end the political chaos and economic uncertainty that Brexit has unleashed.
The assessments left May's deal hanging by a thread.
Tory Brexiteers and the DUP are taking legal advice on the changes but Labour said the PM had secured nothing new.
In a written legal opinion, Cox said that if UK-EU negotiations became stalled through "intractable differences", Britain would have "no internationally lawful means of exiting the Protocol's arrangements, save by agreement".
"The House was clear on the need for legally binding changes to the backstop".
That appeared unlikely after Cox's assessment.
Speaking of the "legally binding changes" the United Kingdom agreed with the EU, Irish Prime Minister Leo Varadkar said earlier Tuesday that Britain had been offered "reassurance and guarantees" over the Brexit deal as MPs in London prepare to vote Tuesday on whether to accept the withdrawal agreement, but denied that the concessions brokered alterations to the so-called Irish backstop.
Supporters of Brexit argue that, while a "no-deal" divorce might bring some short-term instability, in the longer term it would allow the United Kingdom to thrive and forge beneficial trade deals across the world.
Mr Cox looked at the changes that Mrs May had agreed, but concluded that the legal risk had not changed.
Britain's main opposition Labour party confirmed Monday that it would again vote against the deal. "The government's strategy is now in tatters".
Sterling rose on news of the deal but then gave up its gains by mid-morning and was trading $1.3163 at 1040 GMT.
Zampa puts India in a spin as Australia seal stunning series win
Ravindra Jadeja replaced Yuzvendra Chahal in the Indian playing eleven while K L Rahul made way for Mohammad Shami. Some late hitting from Bhuvneshwar Kumar, who struck 46 from 54 balls, could not get India over the line.
However, the motion also notes that if the Commons has not passed a resolution approving the negotiated Withdrawal Agreement by March 20, then it is "highly likely" the European Council would require a "clear goal for any extension" and to determine its length. "There is no alternative".
In addition, there was a joint statement that focused on the future relationship between the European Union and United Kingdom which requires that an alternative to the backstop should be in place by December 2020.
Two documents were agreed after Mrs.
Mrs May began her day by huddling with top team members in preparation for her final attempt in parliament to sway sceptical MP's hearts and minds.
"In politics, sometimes you get a second chance".
The second vote on May's Brexit deal, up until last night, was widely believed to be likely rejected.
Brexit will pitch the world's fifth largest economy into the unknown and many fear it will divide the West as it grapples with both the unconventional US presidency of Donald Trump and growing assertiveness from Russian Federation and China.
Will the DUP back Theresa May's Brexit deal?
Extending the timeframe for Brexit requires approval from all 27 remaining European Union member countries.
DUP sources have said the party can not support the Prime Minister's deal in tonight's vote, which effectively kills off any chance of it getting Commons' backing.
"Today is our Hotel California moment". This is what we can expect tonight.