Theresa May has insisted the government is “getting on” with Brexit, following a High Court ruling that Parliament must vote on when the formal process of leaving the EU can get under way.
The prime minister urged MPs and peers to “remember” the referendum result.
UKIP leader Nigel Farage warned of protests on the streets if the decision in favour of Brexit was ignored.
But the campaigner who brought the High Court case said it would stop ministers acting like a “tin-pot dictatorship”.
Judges ruled on Thursday that Parliament should vote on when the government can trigger Article 50 of the Lisbon Treaty, starting formal negotiations with the EU.
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Mrs May has promised to invoke Article 50 by the end of next March.
The government argues ministers already have sufficient powers – under the Royal Prerogative – to do this without MPs and peers having a vote. It has vowed to fight to get the ruling overturned next month in the Supreme Court.
Health Secretary Jeremy Hunt told BBC One’s The Andrew Marr Show the prime minister had to be allowed “latitude” when negotiating with the EU over Brexit.
He said: “The impact on the economy will be far worse if, through some parliamentary mechanism, Theresa May is forced to lay out her entire negotiating strategy.”
Speaking at Heathrow Airport as she left for a trade mission to India, Mrs May said: “I think we all have to remember, and what MPs and peers have to remember, is that we had a vote on 23 June.
“The British people, the majority of the British people, voted to leave the European Union. The government is now getting on with that.”
She added: “I want to ensure that we get the best possible deal for the UK as we leave the EU, that’s the best possible deal for trading with and operating within the single European market.
“But alongside that, the UK will be a confident, outward-looking nation, taking its place on the world stage, looking to build relationships around the globe.”
The row has escalated in recent days, with several newspapers being highly critical of the judges who made the High Court decision, the Daily Mail branding them “Enemies of the people”.
The government says it will not be put off its Brexit timetable. Under this, the two years of negotiations with the EU are due to end in 2019, when the UK will leave the 28-member organisation.
Also speaking to Andrew Marr, Ms Miller, the investment manager who was the lead claimant in the High Court case against the government, said: “Everyone in this country should be my biggest fan, because we have used our own money to create certainty about the way ahead.”
She added: “Do we want a country where we have no process?”
“The case is that [Mrs May] cannot use something called the Royal Prerogative to do it because we do not live in a tin-pot dictatorship,” Ms Miller said.
‘Spanner in works’
But Mr Farage said the court’s decision meant the country was faced with “half Brexit”, adding that the “reach of the European Union into the upper echelons of this country makes it quite difficult for us to trust the judgement”.
He warned: “If the people of this country think that they’re going to be cheated, they’re going to be betrayed, then we will see political anger, the likes of which none of us in our lifetimes have ever witnessed.”
Asked if there was a danger of disturbances in the street, he replied: “Yes, I think that’s right.”
Mr Farage called for those who were in favour of Brexit to take part in peaceful protests and in any election not support people who wanted to overturn the process.
Meanwhile, Labour deputy leader Tom Watson told BBC 5 Live’s Pienaar’s Politics: “We are not going to hold this up. The British people have spoken and Article 50 will be triggered when it comes to Westminster.”
He added: “Ultimately when the vote comes, Labour will support Theresa May to trigger Article 50.”
It comes after Labour leader Jeremy Corbyn told the Sunday Mirror: “The court has thrown a big spanner in the works by saying Parliament must be consulted. We accept the result of the referendum.
“We are not challenging the referendum. We are not calling for a second referendum.”
But he said Labour would push for the party’s “Brexit bottom lines”, which include access to the European single market.
Liberal Democrat leader Tim Farron said political leaders had a “duty to calm tempers, heal division and work together to keep Britain open, tolerant and united”.
And Scotland’s Brexit minister, Michael Russell, said he could not imagine any circumstances in which SNP MPs would vote in favour of triggering Article 50.
UK PM vows to carry out Brexit ‘in full’