The Democratic Unionist leader acted after his bid to have the Stormont assembly adjourned in the wake of a murder linked to the IRA was defeated.
Mr Robinson said recent developments in the case, including the arrest this week of Sinn Fein’s northern party chairman, Bobby Storey, in connection with the murder, had “pushed devolution to the brink”.
Mr Storey was released without charge last night, hours after Mr Robinson announced he was stepping aside.
Northern Ireland Secretary Theresa Villiers said working relations at the power-sharing executive had suffered a “complete breakdown”.
Ms Villiers said she would not suspend the devolved institutions following the resignation of DUP ministers, including Mr Robinson, but acknowledged the situation was very grave.
“Power-sharing only works effectively if you can have effective relationships between parties from different sides of the community and different parts of the political spectrum,” she said.
Ms Villiers is chairing talks with Irish Foreign Affairs Minister Charlie Flanagan at Stormont involving the five largest Northern Ireland parties.
“I will be working and continuing to work with determination with the Prime Minister, the Northern Ireland parties and colleagues in the Northern Ireland government to get a way through these crises and find a resolution to the hugely important challenges we now face.”
She added: “We did not think the circumstances will justify suspension, that has not changed, and obviously suspension would not resolve the two big problems we face – implementation of the Stormont House Agreement and the presence of paramilitary organisations.
“The only way those problems are going to be resolved are through intensive cross-party talks. I think time is very short because of the seriousness of the situation, I think that today’s events make the situation much more serious.”
Mr Robinson said the current DUP finance minister Arlene Foster would remain in the Executive and take over as acting First Minister.
The mass walkout from the coalition came after the DUP failed to get the Assembly adjourned for a period to allow crisis talks to address the political implications of the murder.
Mr Robinson repeated a demand for the government to suspend the institutions outright to enable space for the talks to happen, but Ms Villiers rejected the call.
The fallout from the murder of Kevin McGuigan has already seen the Ulster Unionists resign their one ministerial post.
The exit of Mr Robinson along with three of the DUP’s four other ministers has left the 13-minister administration in freefall. Collapse of power-sharing is not inevitable but its demise appears to have been hastened by the day of dramatic developments.
The DUP wanted all Assembly business suspended to allow crisis talks to take place about the political consequences of the murder.
Mr Robinson’s announcement came after Sinn Fein, the SDLP and the Ulster Unionists voted against a DUP proposal to adjourn the Assembly.