SINN Féin president Mary Lou McDonald has described Paul Givan's resignation as "one of those defining moments" and said her party would not allow an executive to "stagger on in the months ahead".
Mrs McDonald called for May's assembly election to be brought forward following the First Minister's resignation.
Her remarks came as Mr Givan announced he was quitting some five months after the move was first mooted by Sir Jeffrey Donaldson.
The DUP leader said yesterday he had told Brexit negotiator Liz Truss at the beginning of last month that his deadline for "real progress" in negotiations between the UK and EU was January 31.
He accused the British government of failing to "honour" its commitment to "protect Northern Ireland's place within the UK internal market".
"I warned that as leader of the DUP, I was not prepared to lend my hand to a protocol which so fundamentally undermines the union and the economic integrity of the United Kingdom and Northern Ireland's position in it," he said.
Sir Jeffrey also attacked his opponents at Stormont.
"We've had enough of Sinn Fein, the SDLP and the Alliance Party seeking the rigorous implementation of a protocol which we know is a bad deal for Northern Ireland," he said.
"We've had enough of governments talking about power-sharing but ignoring the fact that every elected unionist in Northern Ireland rejects the Irish Sea border."
Mr Givan said the protocol had undermined power-sharing but he hoped for a "resolution to the issues that have regrettably brought us to this point".
He described serving as first minister as "the privilege of my lifetime".
"When I first entered the assembly 12 years ago, I never expected to have the opportunity to lead the government and serve the people of Northern Ireland as first minister."
But the DUP's decision to pull Mr Givan from the executive – forcing the resignation of his Sinn Féin counterpart Michelle O'Neill – was met with almost universal criticism, with only Jim Allister welcoming the move.
The TUV leader said it was "about time" the DUP acted.
"TUV has long said key to getting rid of the union-dismantling protocol was ending all complicity in its implementation," he said.
Mrs McDonald said the resignation was a "disgrace", coming at a time when crucial legislation needed to be passed.
Speaking in Belfast alongside Ms O'Neill and Sinn Féin Finance Minister Conor Murphy, she said an early election "must be called and the people must have their say".
"This ruse by the DUP to ratchet up tension and pressure in an ill-conceived attempt to get rid of the protocol will not succeed, the protocol is necessary, the changes or adaptations that are required to it, that work will be done through the Joint Committee, but for us ultimately, the people will finally be the arbiters and the decision makers in all of these regards," she said.
Ms O'Neill said she had contacted other party leaders to arrange discussions about outstanding legislation.
"It's very clear there are catastrophic impacts in terms of the DUP's action today and there are many many casualties as a result of their political opportunism," she said.
SDLP leader Colum Eastwood described the DUP leadership's actions as a "gross betrayal" of people from all backgrounds across the north.
“Resigning from government when people are struggling to provide for their families, heat their homes and deal with the cost of living tells you all you need to know about the DUP – for them the party will always come first and ordinary people come last," he said.
Alliance leader Naomi Long said the DUP would " pay the price for this cynical electioneering".
"The people of Northern Ireland gave a resounding message during the last political impasse – they want devolution, with locally-accountable politicians taking decisions in the public interest," she said.
UUP leader Doug Beattie said the DUP had manufactured a crisis.
"It is time for unionism to be led by those who know how to work with others to get the best for the people of Northern Ireland,"he said.
Secretary of State Brandon Lewis said yesterday's developments were "extremely disappointing".
"I urge them to reinstate the first minister immediately to ensure the necessary delivery of public services for the citizens of Northern Ireland," he said.
