Brexit-backing MPs reacted with anger to the comments, accusing Mr Tusk of "arrogance".
Britain's Brexiteers with no plan of how to deliver deserve a "special place in hell", the EU's Donald Tusk said, adding he no longer believed there was a way to stop Britain leaving due to the "pro-Brexit stance" of both the prime minister and the leader of the opposition.
Tusk said the Irish border issue and the need to preserve the peace process remained the EU's "top priority".
REUTERS/Yves Herman Taoiseach Leo Varadkar talks to European Council President Donald Tusk during their meeting at the Europa building in Brussels, Belgium, February 6, 2019. And this is why we insist on the backstop.
Tusk's apparently pre-planned remarks, which were tweeted from his account on the social media website as he spoke the words at the press conference, failed to impress Brexit-supporting politicians in Britain, where the message provoked some ire.
"We will not gamble with peace or put a sell-by date on reconciliation, and this is why we insist on the backstop", Tusk said.
In her efforts to break the impasse over her Brexit deal, May signalled she will seek changes rather than outright removing the backstop, which is created to preserve the open border between Northern Ireland and European Union member state Ireland.
Yesterday, Mrs May suggested she is seeking "changes" to the backstop in her Brexit deal, rather than its total removal from the Withdrawal Agreement.
Britain, Ireland and the European Union want to avoid physical checks on the border between Ireland and Northern Ireland that ceased with the 1998 Good Friday Agreement.
The EU will not renegotiate the divorce deal and the Irish backstop but alternatives could be worked on after Britain leaves, chief EU negotiator Michel Barnier said on Monday.
During a speech Tuesday in Belfast, Northern Ireland, May restated her "unshakeable" commitment to avoiding a hard border and said she didn't plan to remove the "insurance policy" entirely.
British ministers, The Sun newspaper said, have been examining a plan drawn up by Japan's Fujitsu to track trade across the border, while the Telegraph said ministers had discussed delaying Brexit by eight weeks.