The SDLP has refused to replace Nichola Mallon as Infrastructure Minister.
Colum Eastwood said his party will not nominate a replacement to sit in a "zombie Executive".
The party leader said it would be undemocratic for his party - which was returned with a reduced mandate - to be in the Executive.
There is confusion as to if another minister can be appointed without a first and deputy first minister. However Stormont sources have indicated it is possible and the position would go to Sinn Fein under Stormont rules.
"From our perspective, we're now in opposition, we will act constructively, work with other parties but we will also hold ministers to account where we have to," he said.
"We don't have a mandate for government, the people have spoken. I respect what the people have said, they've made it very very clear.
"We don't have enough seats in the Assembly to form a government. I don't feel good about that but I'm humbled about it.
"We can't be in government, it would be totally undemocratic for us to sit in a zombie Executive and have no power to do anything when we haven't even got a mandate from the people.
"We know who got the mandates, we will support them when we have to, we will also hold them to account, but they should get on with it today and form a government."
While talks are ongoing on forming a new Executive following the result of Thursday's election, ministers from the last mandate remain in place. They can still remain in their positions for at least six months if no Executive is formed.
The SDLP deputy leader lost her seat in north Belfast amid a surge in votes for the Alliance party over the weekend.
Ms Mallon was excluded in the final stage of the count on Saturday evening, when former Belfast lord mayor Nuala McAllister of Alliance filled the the fifth and final seat.Colum Eastwood also called on the DUP to appoint a deputy first minister as a matter of "urgency".
He said: "What is absolutely clear, is that a government needs to be formed urgently...so I call on Jeffrey Donaldson to commit to appointing a deputy first minister right away."
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