He further noted that Britain's "backstop" proposal for avoiding controls at the Irish border raises more questions than answers, warning that Brussels would never accept an end-date being included in the plan.
MPs could still vote down any Brexit deal but wouldn't be able to direct what the government does next.
And he said Mrs May's plans for a temporary backstop were not in line with the needs of Brussels, Dublin or Northern Ireland for stability, turning one of her own slogans against her to declare: "Backstop means backstop".
The document published by the United Kingdom states that the backstop option "will be only in place until the future customs arrangement can be introduced" and that "the United Kingdom expects" this to happen by the end of December 2021.
"It would tie us effectively into the EU's customs arrangement for an indefinite period", he said.
Under its "backstop" proposal, there would be no tariffs, quotas, rules of origin or customs declarations on all UK-EU trade, not just trade between Northern Ireland and the Republic of Ireland.
And the paper adds a temporary arrangement "could only be provided in law" if an EU-UK divorce deal and a future partnership framework are agreed.
Mr Barnier said he continues to have "great respect" for the prime minister.More news: LeBron James ‘played with broken hand’ in last 3 NBA Finals games
"How does that fit with the need to secure the absence of a hard border in all circumstances?"
Earlier, Irish Prime Minister Leo Varadkar said he could not agree to a time-limited version of the backstop arrangement.
Launching the roadmap in Westminster, Best For Britain chief executive Eloise Todd said: "A people's vote will be held before the end of March 2019".
Another option would be to remove Northern Ireland from the customs union without a hard border, instead having very light border checks or even none at all.
Dr Fox later told the BBC that the new proposal ensured that Britain would be "leaving the European Union and any implementation and subsequent arrangements by the end of 2021". We believe this is the right thing to do so that everyone's voice can be heard.
"This was a hard decision to take because it involved the balance of a number of different issues".More news: EA Origin Access Premier subscription gets pre-launch games
Davis has not commented directly on the source-based reports of his anger.
Tory MP Nadine Dorries had also ramped up the political drama by citing Mr Davis's SAS training, as she warned he is "trained to survive" but also "trained to take people out".
On Thursday, Theresa May was forced to say she "expects" there to be a cut-off date after Brexit Secretary David Davis allegedly threatened to resign.
Shadow Brexit secretary Sir Keir Starmer described the PM's concession as "embarrassing".
Mr Barnier said it would not be feasible to extend the European Commission's backstop plan - under which Northern Ireland would remain part of the EU's customs territory after Brexit - to include the whole of the UK.
Prime Minister Theresa May declined to say Wednesday when the government would publish a promised document setting out its negotiating position.
A campaign for a another European Union referendum - this time on the final Brexit deal - has been launched in London.More news: Disturbing nursery rhyme teaches kindergartners how to survive a school shooting