Phase one of the Brexit negotiations ended with a last-minute deal signed between Prime Minister Theresa May and EU boss Jean-Claude Juncker.
But Ms Gill said the Brexit Secretary threatened the talks because of his "loose lips".
The Labour MEP said: "In another new chapter of the never-ending story ‘David Davis vs. Reality’, we saw the Brexit minister scrambling to mend the relationship with Brussels after his loose lips nearly undermined everything that had been agreed.
NEENA GILL/PARLIAMENT TVBrexit news: Neena Gill claimed David Davis risked to "undermine" talks with the EU
"Maybe Davis was attempting to divert attention from his lies to the House of Commons about doing an impact assessment of Brexit for vital sectors in the UK. This deal means more smoke and mirrors."
Ms Gill also suggested the British Government had reverted to "ambiguous words" to confuse the public about the terms of the Brexit deal reached with the European Union.
She continued: "Davis has claimed that a future trade deal would be on the table the day after Brexit was dismissed by the chief negotiator Michel Barnier.
"The Government is sticking to ambiguous words