Senior Conservatives have called for the party to pull together after it suffered its worst results in English local elections since 1995.
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The Conservatives lost 1,334 councillors in Thursday's votes.
Home Secretary Sajid Javid admitted voters had "issues of trust" over Brexit, and said the European elections would "be even more challenging".
But, in a rallying cry to Conservatives in Aberdeen, he said that "a divided party cannot unite a divided nation".
'Face facts'
Meanwhile, Health Secretary Matt Hancock will tell Conservatives in Wales to "pull together" in a speech later.
And, in an interview with the Daily Telegraph, Environment Secretary Michael Gove has urged Eurosceptic MPs to "face facts" and accept that Parliament will not approve a no-deal Brexit.
Justice Secretary David Gauke told BBC News that the local election results should be seen as a "punishment" to both the Conservatives and the Labour party "for failing to find a way through" the Brexit conundrum.
"What we need to be doing is addressing the big issue that is in front of us, which is Brexit," he told BBC Breakfast.
The MP for Hertfordshire South West rejected calls to oust Theresa May, adding: "We should back the prime minister and try to find a way through this situation, so that we can bring the country together again, we can unite the Conservative party, and find a practical way through."
The UK was due to leave the EU on 29 March, but the deadline has been pushed back to 31 October.
Mr Javid said that was a big factor in the Conservatives losing control of 45 councils on Thursday, in its worst performance since John Major's party lost 2,000 councillors in 1995.
The home secretary said the party risked losing voters' trust after "not delivering on a promise at the heart of our last manifesto".
And, speaking about the European elections, due to take place on 23 May, he said: "We shouldn't be surprised if people tick the protest box on the ballot paper.
"Without anything else at stake, it will be a verdict on the delivery of Brexit."
In his speech, Mr Hancock will say the party must "deliver Brexit" but not be "defined by Brexit." And Mr Gove said it was time for MPs on all sides to make Brexit happen.
"Labour MPs will be reflecting this morning and over the weekend on the message that their voters are telling them, which is we voted to Leave," he told the Telegraph.
"I hope they will recognise that they need to work with the Government in order to deliver Brexit."
In the local elections, the Liberal Democrats - who have campaigned for a second vote on leaving the EU - were the main beneficiary of Tory losses, gaining 703 seats.
Lib Dem leader Sir Vince Cable said every vote for his party was "a vote for stopping Brexit".
Labour lost 82 seats - having also suffered losses the last time these council seats were contested, in 2015.
The Green Party added 194 councillors, while the number of independent councillors rose by 612.
The BBC projects that, if the local election results it analysed were replicated across Britain, both the Conservatives and Labour would get 28% of the total vote.
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- Local elections: The main parties punished
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- May predicts 'bright future' after Brexit
Polling expert Prof Sir John Curtice said the days of the Conservatives and Labour dominating, as happened in the 2017 election when they won 80% of the vote between them, "may be over".
He said it was only the second time in history that the two main parties' projected national share of the vote had fallen below 30%.
The only other occasion was in 2013, when UKIP performed strongly in local elections.
