The move will stop the devolved Assembly from being able to function.
Earlier, DUP Leader Sir Jeffrey Donaldson confirmed his party will not nominate a speaker as part of its opposition to the Northern Ireland Protocol.
Without a speaker, the Assembly will be prevented from carrying out its business.
Sinn Fein Stormont leader Michelle O’Neill said the DUP has “punished the electorate” by boycotting the election of a speaker of the Northern Ireland Assembly, and that “the public deserve better”.
Alliance Party leader Naomi Long branded it a "sad day for Northern Ireland" and a "shameful day for the DUP."
Speaking to reporters in the Great Hall after the Assembly was adjourned, Ms O’Neill said the DUP’s action to boycott the election of an Assembly speaker “isn’t tolerable, it isn’t acceptable, it isn’t good enough”.
Speaking in the Great Hall at Stormont, Mrs Long added: “Despite the fact that the vast majority of people in Northern Ireland voted for parties that wanted to return to government, that wanted to see the Assembly work, and despite the fact that even those who voted for the DUP gave them no mandate to block a return to the Assembly, we have found ourselves in that situation today.
“But while this is a sad day for the people of Northern Ireland, it is a shameful day for the DUP.
“The day the DUP came to Stormont, signed the register, took their salaries but refused to take their seats and do the work to earn it.
“I don’t think that is ever acceptable but it is particularly unacceptable when people in our constituencies are struggling to feed their families, struggling to heat their homes, worried for their futures and it puts all of us as politicians in a place which is embarrassing once again.
“We want to serve the public but are prevented from doing so.”
Ms Long continued: “I am appalled that we could not even challenge the person who made this decision (Sir Jeffrey Donaldson) because having been returned as an MLA only a week ago, he has disappeared off to his safety net at Parliament.
“Despite all of that, we are here ready to work. We might not be able to go into the chamber and do the work as we intended but we will be in our constituencies working for the people who elected us.
“I will be in my ministerial office working for the people who elected me and we will continue to do all that we can to make life better for the people of Northern Ireland irrespective of those who have nothing to offer but more drama, more division and more disruption.
“It is now time for them to call this to a close, allow us to get a speaker, allow us to move on and allow Northern Ireland to move forward.”
Also today, Ms O’Neill announced that Sinn Fein MLA for Upper Bann John O’Dowd would be taking up the role of Infrastructure Minister in a caretaker capacity, after the former minister Nichola Mallon of the SDLP lost her seat in last week’s election.
What happened in today's plenary hearing
Newly-elected MLAs entered the Assembly chamber on Friday for the first plenary sitting since the election.
Ms O'Neill had told MLAs the public is hoping that Northern Ireland's elected parties have "the maturity and courage" to take responsibility, adding that "there is absolutely no reason we should be in a rolling crisis, even for one second".
It is the job of politicians to "properly fund" the healthcare service and to agree a three-year budget and invest in the health service, Ms O'Neill said.
"This is our hour of decision, not tomorrow, and not for a moment longer can the DUP deny democracy, punish the public, boycott this Assembly and executive, and prevent us from putting money in people's pockets.
"Every one party in this chamber told the electorate that they would turn up on day one. Well, the DUP have failed on day one."
Earlier, DUP MLA Paul Givan had told the Stormont Assembly that his party would not be supporting the election of a speaker.
The Ulster Unionists nominated Mike Nesbitt and the SDLP nominated Patsy McGlone.
Mr Givan told MLAs: "The DUP received a mandate to remove the Irish Sea border and our mandate will be given respect. Our message is now clear, it is time for action, words will no longer suffice.
"It is because we want these institutions to endure that we are taking the action we are taking today.
"Northern Ireland works best when we work together. Those who now call for majority rule need to recommit themselves to the principles of the Good Friday Agreement.
"We will not be dictated to, we will be treated with respect and equality. Now is the time for action."
Naomi Long used her Assembly speech to address the challenges facing politics in Northern Ireland.
The Alliance Party leader said: "We are here today in order to elect a speaker so the Assembly can go about its business so that those who have been elected can serve the people who elected them.
"We come here with a can-do attitude and a commitment to serve the people who elected us.
"Many of us in this chamber represent people who did not consent to Brexit in the first place. And yet we turned up for work.
"We also don't all have equality. Some in this chamber are more equal than others and myself and my 16 colleagues' votes will count for less in this next election than everyone else in this chamber. So if we're really committed to equality, we will also be committed to reform of these institutions.
"To turn up here, to sign in, to take salaries and to refuse to take seats is a slap in the face for every family that struggles to make ends meet, for every person who sits on a waiting list.
"I would appeal to the DUP to think long and hard before they insult the electorate by doing so today."
UUP leader Doug Beattie urged that an Assembly speaker be elected so that the public's concerns can be addressed.
After standing in silence before MLAs for several seconds, Mr Beattie said: "Silence. The same silence we were subjected to for three years when Sinn Fein walked out. The same silence we're now going to be subjected to if the DUP don't support a speaker."
"People will go cold and hungry in their homes, and from this place there will be silence," he said.
"We can today make the point in regards to the protocol, but also elect a speaker in order to do some business so we don't have silence."
SDLP MLA Matthew O'Toole told the Assembly that he was happy to nominate party colleague Patsy McGlone as speaker.
But he said: "All of my words are clearly in vain because the DUP has decided to thwart democracy.
"They are also demeaning democracy.
"In stifling the creation of an executive and the election of a speaker, the DUP has demeaned the entire democratic process. Shame on them."
Traditional Unionist Voice leader Jim Allister said it was not appropriate to nominate a speaker to the Assembly while the Northern Ireland Protocol is in place
Mr Allister, the only MLA returned for his party in the election, said: "Since the leverage is in respect of this Assembly, that is why the mendacious Prime Minister we have has to be brought to the point of choice. Does he want to save the protocol or does he want to save these institutions?
"Until this protocol becomes moribund, then this Assembly must be moribund."
People Before Profit MLA for Belfast West Gerry Carroll criticised the DUP for their "obstruction" to electing a new Assembly speaker as communities struggle with rapidly rising energy and fuel bills.
"For all their talk about the protocol, poll after poll has shown that it isn't a priority or the number one issue that people are vexed about," he told the Stormont Assembly.
"In many ways, this is a manufactured crisis.
"The vast majority of people did not vote for this and should not be held to ransom by the DUP's self-serving actions."
The Secretary of State for Northern Ireland has said it is "disappointing" that a new speaker has not been nominated for the Assembly.
"Great to have MLAs back in Stormont today, but disappointing to see a Speaker has not yet been nominated," Brandon Lewis tweeted.
"The people of Northern Ireland voted and deserve a stable and accountable devolved government. I urge the parties to come together and form an Executive."
Meanwhile, outgoing speaker Alex Maskey thanked his Assembly colleagues, as well as his family, in a speech to the Assembly.
He also told MLAs that politicians in Northern Ireland had come through political difficulties before.
Mr Maskey said: "I recognise that we are currently in a difficult political situation.
"Since 1998, we have all seen our fair share of those. Those of us who were here in 1998 and since then had big issues to deal with. However we did come through them.
"The last two years we were able to meet the challenges of getting the Assembly re-established and keeping the Assembly functioning to take important decisions during the pandemic. "
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