A sitting of the Northern Ireland Assembly to nominate new leading ministers has been axed after negotiations to form a new coalition collapsed.
Northern Ireland Secretary of State James Brokenshire has said a "short window of opportunity" exists to restore a powersharing executive in the region.
As the deadline to form a new Stormont administration passed at 4pm on Monday, following last month's snap elections, Mr Brokenshire said there remained "an overwhelming desire" for strong and stable devolved government.
"We now have a short window of opportunity to resolve outstanding issues and for an executive to be formed," he said at a press conference outside Stormont House.
"Everyone owes it to the people of Northern Ireland to grasp that and provide the political leadership and stability that they want."
Mr Brokenshire said he would make a full statement in the House of Commons on Tuesday setting out a way forward for the region.
DUP leader Arlene Foster claimed Sinn Fein's "inflexible" approach to negotiations was to blame.
She said she did not believe another election would solve anything.
"We wonder whether Sinn Fein were serious about reaching agreement at this time," said the former first minister.
"We are just disappointed that Sinn Fein did not come to the talks in the same spirit as we came to the talks.
"We respect everybody's mandates, let me make that very clear, but if we wanted to form an executive, then there had to be a spirit of compromise and unfortunately that didn't exist.
"The government of Northern Ireland is not a game, it is actually very serious and the fact we do not have an executive being formed today is very regrettable."
Amid the trading of recriminations, Mrs O'Neill offered a very different view on culpability.
She claimed the DUP had failed to live up to previous agreements and were standing in the way of progressive policies.
"We are standing firm - previous agreements need to be implemented," she said.
"We came at the negotiations with the right attitude, wanting to make the institutions work, wanting to deliver for all citizens.
"Unfortunately, the DUP maintained their position in relation to blocking equality, delivery of equality for citizens - that was the problem."
Sinn Fein has said it will not share power with Mrs Foster as first minister until a public inquiry into the Renewable Heat Incentive (RHI) scheme is concluded.
Republicans have also been seeking movement on issues such as an Irish language act giving the tongue official status in Northern Ireland, a hugely symbolic measure but deeply problematic for some unionists.
A voting surge by Sinn Fein in the snap Assembly election earlier this month saw the party come within one seat of becoming the biggest party at Stormont behind the DUP.
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