The Tanaiste said that while he welcomed Mrs May's vow not to allow Brexit to jeopardise the Northern Ireland peace process, the United Kingdom must now provide the solution to how a hard border could be avoided and the Good Friday Agreement protected.
And she declined to defend Boris Johnson's comparison of the border to crossing between London congestion zones in Camden and Islington, but insisted both of them are "absolutely clear" that there will not be a hard border.
Also speaking to the BBC on Sunday, the Prime Minister said her overall Brexit plan for Britain to align with European Union rules to allow for "frictionless" trade while retaining the leeway to diverge from them was "ambitious" and "credible".
In a highly complex, yet-to-be-worked-out plan, Mrs May essentially said the customs partnership could keep the same border tariffs for goods intended for the EU.
Writing in the Telegraph, he continued: "Moreover, whatever sectors a customs union covered, were the EU to make its own new trade agreements in the future, we would be forced to allow goods from other countries into our market tariff-free, on terms set by Brussels, without any guaranteed tariff-free access to the markets of other countries in return".
In the past, former Taosieach Bertie Ahern said this would require a "blind eye".
David Lidington, who is Theresa May's unofficial deputy, said Westminster would only get involved in devolved areas if a "pause" is needed to make UK-wide frameworks to protect the British common market and global rules.
"When someone definitively says something will be the case from the British government, people assume that is the negotiated outcome".
And she called for "mutual recognition" in areas like regulation of goods, broadcasting licences and professional qualifications.
Missing snowboarder found deceased at Squaw Valley
On Friday, another snowboarder, Heather Turping, 39, heard someone scream "Avalanche!" as she saw a "cloud of snow coming down". We hope for the full and speedy recovery of the injured guests, and extend our thoughts and support to all those involved.
Britain's Prime Minister Theresa May delivers a speech about her vision for Brexit, at Mansion House in London, Britain, March 2, 2018.
Nicola Sturgeon has raised the possibility of delaying her decision on another independence referendum. She wants recognition of broadcasting rules to allow United Kingdom channels to continue to be watched across all European countries, and continuity of rail, maritime and aviation services - so that United Kingdom airports don't grind to a halt.
"We would, of course, accept that this would mean abiding by the rules of those agencies and making an appropriate financial contribution", May said.
But while he welcomed May' bid to secure a very close relationship with the European Union after Brexit, that London needed to provide "more detailed and realistic proposals". "Brexit is due to happen in a little over 12 months, so time is short", he said.
"I remain concerned that some of the constraints of leaving the customs union and the single market are still not fully recognised", he said. "The absolute arbiter of disputes over our future partnership cannot be the courts of either party", said the British Prime Minister.
Brexit Secretary David Davis is concerned that approach might tie parts of Britain's economy to the European Union without having a say, with one Cabinet source telling the Telegraph: "It would make us a rule taker".
"We need to resolve the tensions between some of our key objectives", May said in the speech, which was moved from the Brexit-leaning North East of England.
"After what I heard today, I am even more concerned".
- Bound gas pipeline opens in Afghanistan
- Iraqi court sentences 16 to death over Daesh links
- UA outside counsel releases statement on Deandre Ayton
- YouTube stopped hiring white men in attempt to boost diversity, lawsuit claims
- New US Tax Law Brings Warren Buffett A Nice Cheque: $29 Billion
- Dana White Continues To Tease Brock Lesnar's Potential UFC Return
- Maintenance Leads to Drop In UAE's Oil Output
- 7.5 magnitude earthquake strikes Papua New Guinea
- Proposal to change Chinese president's term limits
- Mark Carney: Cryptocurrency isn't real money and must be regulated