The move will stop the devolved Assembly from being able to function.
UUP MLA Mike Nesbitt, one of the candidates for Speaker, was not elected in the cross-community vote.
His candidacy was backed by 51.9% of MLAs, but failed due to a lack of cross-community support.
SDLP MLA Patsy McGlone also failed in a bid to be Speaker, receiving 71.3% of the vote but also failing to receive sufficient cross-community support.
“I believe that we need to send a very clear message to the European Union and to our government that we are serious about getting this protocol sorted out,” he said.
But Alliance Party leader Naomi Long said it had been a “shameful day” for the DUP.
Speaking in the Great Hall at Stormont, Mrs Long said: “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.”
Mrs 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.”
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”.
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”.
Ms O’Neill also 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.