The DUP leader accused Sinn Féin of trying to distract public attention from the issue, adding "this tired tactic does nothing to solve the problem most likely to bring down the political institutions".
The two parties had been at loggerheads over welfare reform cuts that have been introduced in the rest of the UK.
The DUP Finance Minister Simon Hamilton has warned that failure to agree on welfare reform means that Stormont faces an £87m penalty.
Budget adjustments in the June Monitoring Round were also delayed because of a row over putting money aside to account for this.
Mr Robinson responded after Sinn Féin president Gerry Adams warned that the political process is in trouble.
He said: "Over the past week we have heard from a series of Ministers of the first set of consequences resulting from the Sinn Féin imposed cuts arising from its refusal to accept the welfare deal negotiated for Northern Ireland.
"The deal negotiated would have provided the best welfare payment system in the UK. It ditches the bedroom tax for all our existing tenants and provides tens of millions of pounds in a contingency fund for those most impacted by the new Coalition Government's Welfare Reform arrangements.
"Public spending on welfare across the UK will continue to rise, albeit at a slower rate."
He added: "No responsible party would allow the reductions in departmental funding to happen as a consequence of failing to accept the negotiated arrangements."
By far the most damaging issue that has the potential to end devolution is the shameless denial by Sinn Féin of economic realities resulting from welfare reform.
DUP leader Peter Robinson
However, Sinn Féin MP Conor Murphy said the First Minister "continues to duck responsibility for the growing difficulties in the political process".
"His pandering to anti-agreement elements and his support for anti-working class Tory cuts are clear evidence that, under Peter Robinson's leadership, unionism is moving steadily to the right," he commented.
"Sinn Féin for our part makes no apology for opposing vicious Tory welfare cuts."
Earlier Gerry Adams blamed the so-called anti-Good Friday Agreement axis within unionism, the pro-unionist stance of the NI Secretary of State, and the refusal of Downing Street to own its obligation for the region's problems.
"Most worryingly there is no evidence from Downing Street or the NIO or the Unionist leaderships of any likelihood of a real negotiation on all of these issues commencing in September," he said.
Mr Adams said the Tory-led government wants to impose the changes in Northern Ireland, which he said has resulted in disastrous consequences for the disabled, the unemployed and those in low-paid jobs in England, Scotland and Wales.
These changes are not about reform. They are about cuts and they are part of a Thatcherite agenda designed to dismantle the welfare state. And Sinn Féin will oppose them.
Sinn Féin president Gerry Adams
UUP leader Mike Nesbitt accused Mr Adams of "blustering" and his party of not engaging in "politics of the real world".
He said: "John O'Dowd is now saying that he has no idea what the implications of June Monitoring are for other Executive Ministers and Departments, so the obvious question is, just why on earth did he vote for it?
"In admitting he voted for something without being aware of the implications he was reckless in the extreme."
SDLP MLA Alex Attwood said that his party agrees that politics across a range of issues was "degrading" but accused Sinn Féin of "revisionism".
"The comments of Gerry Adams seek to downplay the failures of the republican movement after 1998 by asserting that our latest political challenges are the greatest since 1998. This is political revisionism. It is self-serving and full of self-regard," he said.
"The reasons we are facing immense challenges are many. Party political unionism has moved to the right, with little evidence that it will come back to the middle ground soon."