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 could trigger Article 50 of the Lisbon Treaty.
Mrs May has promised to get this done by the end of next March.
The government, which argues ministers already have the powers to trigger Article 50 without MPs and peers having a vote, has vowed to fight to get the ruling overturned next month in the Supreme Court.
Labour leader Jeremy Corbyn said his party would block the triggering of Article 50 in Parliament if Mrs May did not guarantee continued access to European the single market.
Speaking on BBC One's Andrew Marr Show, Gina Miller, the investment manager who brought 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?"
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 said: "The temperature of this is very, very high. I'm going to say to everyone who was on the Brexit side, 'Let's try and get even. Let's have peaceful protests and let's make sure, in any form of election, we don't support people who want to overturn this process.'"
The row has escalated in recent days, with several newspapers being highly critical of the judges who made the decision, the Daily Mail branding them "Enemies of the people".
Mrs May insists the government 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.
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."
'Latitude'
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."
Health Secretary Jeremy Hunt told The Andrew Marr Show said Mrs May 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."
It was, anyway, "highly unlikely that Parliament would not, in the end, back a decision to trigger Article 50", Mr Hunt said.
'Spanner in works'
However, Jeremy Corbyn said Labour would block the prime minister from triggering Article 50 unless she agreed to the party's "Brexit bottom line", which includes access to the European single market.
He 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. We're calling for market access for British industry to Europe."
Mr Corbyn said the opposition would not allow Article 50 to go ahead unless Mrs May agreed four principles:
access to the single market
a commitment to EU workplace rights
guarantees on safeguarding consumers and the environment
A pledge to commit funds for any EU capital investment lost by Brexit
The Labour leader said his party "would be ready" if the government decided to call an early election.
But Mr Hunt said: "I think a general election is, frankly, the last thing the government wants.
"Theresa May wants to get on with the job and frankly it is the last thing the British people want, with all these very, very important national decisions."
BBC News.
