Key Scottish and Irish parties reacted with fury on Tuesday after Conservative Prime Minister Theresa May said a non-binding British parliamentary vote had empowered her to renegotiate a Brexit backstop protocol that protects an open Irish border.

"Scotland has been silenced, silenced and shafted by the Tories," Ian Blackford, the Scottish National Party (SNP) leader in the British parliament, told lawmakers following the vote.

SNP leader Nicola Sturgeon, who heads Scotland's devolved government, tweeted that lawmakers in the British parliament's main elected house, the Commons, had "indulged [May's] decision to chase a fairy tale... and increased the risk of no deal in the process."

Blackford said May had also "ripped apart the [1998] Good Friday Agreement," also known as the Belfast Agreement, which ended decades of sectarian violence in Northern Ireland.

"In its desperation, the British government is trying to threaten and intimidate the Irish people with a hard border and an economic cliff-edge," Chris Hazzard, a lawmaker from Irish republican party Sinn Fein, said in a statement after the vote.

"The Irish government and EU must stand firm to protect the backstop and Irish interests," Sinn Fein's Brexit spokesman David Cullinane, a member of the Irish parliament, said in a separate statement.

Irish Foreign Minister Simon Coveney reiterated statements by EU officials that the Brexit withdrawal agreement, including the backstop, is not open for renegotiation.

The backstop was agreed by Britain and the EU "as the insurance policy to avoid a hard border [on he island of Ireland] in all scenarios," Coveney tweeted.

"We hope it will never be used, or be replaced quickly by a future relationship agreement," he added. "But it is necessary and tonight's developments at Westminster (London) do nothing to change this."

During the debate before the vote, Blackford said Scotland, where a majority voted to remain in the EU in the 2016 Brexit referendum, was "being dragged out of the European Union against our will."

"Let me be very clear: Scotland must no longer be left at the mercy of events," he said. "Scotland will and must have the right to determine its own future and to choose to be an independent nation within the European Union."

Northern Ireland will leave the EU with the rest of the United Kingdom on March 29, while Ireland will remain an EU member state, making the Irish border the only land frontier between the UK and the EU.

Britain and the EU agreed to use a last-resort backstop to avoid a hard border by keeping regulatory alignment if no other agreement can be found on post-Brexit customs arrangements.

Some politicians in Northern Ireland fear that violence could return if a hard border is imposed after Brexit, potentially undermining the Good Friday agreement.