The Committee on Standards and Privileges said an investigation would be conducted to see if any breaches of the assembly's code had occurred.
Its chairman Declan O'Loan said: "It is clearly in the public interest that an official investigation is carried out in order to establish the full facts in relation to the issues raised in the BBC's recent Spotlight programme."
Earlier, the DUP assembly team said it offered Mr Robinson its "wholehearted support".
Mr Robinson confirmed that he had asked both the House of Commons and the Northern Ireland Assembly to carry out an inquiry into his conduct.
He added: "Throughout this period I will continue to work on the outstanding issues relating to policing and justice and some other matters.
"This allows a particularly concentrated focus on these discussions and we will work to try to build on the agreements we have already reached so that a successful resolution can be reached.
"This is the year to deliver at Stormont for all the people of Northern Ireland."
Mr Robinson's temporary replacement as first minister, Arlene Foster, addressed the Assembly on Monday afternoon.
She said she had already spoken to the Deputy First Minister, Sinn Fein's Martin McGuinness, about how the arrangements would work in Mr Robinson's absence.
The clock is ticking. There is a necessity for all the people in the north to have policing and justice sorted out.
"On behalf of the first minister, I want to make it clear that he entirely rejects this false allegation made by the BBC Spotlight programme and will be seeking to clear his name in the days that lie ahead," she said.
"The departmental solicitor's office has already considered the allegation and has advised Peter Robinson that he was not in breach of the ministerial code, the pledge of office, the ministerial code of conduct or the seven principles of public life."
The Prime Minister Gordon Brown spoke with Arlene Foster on Monday evening.
He said: "I urge all politicians in Northern Ireland, whatever the turbulence of recent events, to remain focused on the business of government, and to recognise the crucial importance of intensifying engagement in those issues which remain to be solved."
The Irish Prime Minister Brian Cowen said he wished Mrs Foster well.
He added: "I want to pass on my best wishes to Peter Robinson and his family as they deal with very difficult personal issues at this time.
"I welcome his decision to continue to work on the crucial issue of devolution of policing and justice and his commitment to a successful resolution.
Mr Dodds, the party's deputy leader, who had been tipped to take over as leader, earlier said that despite attempts by elements of the press and the DUP's political opponents to force Mr Robinson from office, the party was firmly behind him.
He said they wanted him to express their "desire for him to remain in office as leader of the DUP".
Mr Dodds said that at a combined meeting of DUP members of the NI Assembly, party officers, the parliamentary party and their European member, it was agreed unanimously to offer Mr Robinson their full support.
Iris Robinson, an MP and an NI Assembly member, is currently receiving "acute psychiatric treatment" following a BBC Spotlight documentary broadcast last week.
It reported that she obtained £50,000 from two property developers for her 19-year-old lover, Kirk McCambley, to launch a cafe.
She failed to register the money with the authorities in Stormont and Westminster, as required by the law.
It also reported that her husband, the first minister, knew about her financial dealings but did not tell the proper authorities.
It has been announced that she will step down as an assembly member and MP this week due to ill health. Sources have also confirmed that she will be leaving the DUP.
DUP deputy leader Nigel Dodds earlier made a statement in support of Mr Robinson
The DUP and its main coalition partner Sinn Fein have been at loggerheads in recent months over the devolution of policing and justice powers from Westminster to the Assembly in Belfast.
Sinn Fein wants the powers transferred immediately while the DUP has said there needs to be greater community confidence before devolution can happen.
Speaking after Mr Robinson said he would step aside, Sinn Fein's Gerry Kelly acknowledged that Mr Robinson deserved time to sort out personal relationships.
However, he said the political crisis over the devolution of policing and justice was bigger than any one person and must take precedent.
"I don't want this to go on for six weeks, I want this issue sorted out," he said.
"We want to sort it out in the very short term."
The Ulster Unionist Party, the DUP's main unionist rivals, said that Mr Robinson has weakened public confidence in devolution.
Its leader Sir Reg Empey said: "Peter Robinson has indicated the enormous personal pressure he has been under since last March and, putting it bluntly, that sort of pressure cannot be conducive to constructive decision making.
"However, his decision to appoint Arlene Foster as acting first minister does nothing to address the crisis in public confidence in devolution - in fact, it intensifies it.
"At this crucial time in the political process, we now have two lame duck first ministers, neither of whom can make the decisions necessary to ensure proper, working devolution.
"In a highly cynical fashion the DUP have decided to put party before country."
Jim Allister, the leader of Traditional Unionist Voice, which is opposed to the DUP sharing power with Sinn Fein, said the DUP backing for Mr Robinson showed it was out of touch.
He said: "If the DUP is genuinely behind Mr Robinson, then they are more out of touch with public opinion than I appreciated.
"The damage done to the Robinson brand is transferred to the DUP brand by such unequivocal support."
SDLP leader Mark Durkan said Peter Robinson could not come back as first minister's without being "very diminished."
He also suggested that Martin McGuinness, the deputy first minister, was now in a "political twilight zone".
The evening before the Spotlight documentary was broadcast, Mr Robinson revealed that his wife had attempted suicide following the affair.
On Friday, as the storm broke surrounding the allegations, the first minister said he believed he had not done anything wrong.
He has asked government lawyers to investigate his conduct.
He said he would "not be slow" in taking a decision if that investigation found that he should have acted differently.
Over the weekend, a number of parties cast doubt on whether the legal opinion sought by Mr Robinson would have sufficient authority to clear the matter up.
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