Arlene Foster has released a video to Facebook in which she claims Sinn Féin's "actions are not principled they are political"
Martin McGuinness has resigned as Deputy First Minister of Northern Ireland.
He announced his intention to step down earlier on Monday.
It comes amid a scandal around the Renewable Heating Incentive (RHI) scheme, which was designed to encourage businesses to replace older heating sources with more eco-friendly alternatives.
First Minister Arlene Foster has refused to stand aside during an investigation into the scheme.
Mrs Foster, who was elected first minister a year ago, has been under pressure over the botched scheme - which could cost taxpayers there more than stg£400m (€539m).
A lack of cost controls meant businesses were receiving more in subsidies than they were paying for renewable fuel, and the scheme was drastically oversubscribed.
In a statement to the media, Mr McGuinness said: "Sinn Féin will not tolerate the arrogance of Arlene Foster and the DUP.
"Sinn Féin wants equality and respect for all.
"That is what this process must be about. Today I tendered my resignation.
"Today is the right time to call a halt to the DUP’s arrogance. There will be no return to the status quo."
Mrs Foster this evening released a video to her Facebook page in which she said she relays her disappointment that "Martin McGuinness has chosen to take the position he has today."
"His actions have meant that, at precisely the time we need our government to be active, we will have no government and no obvious way to resolve the RHI problems," she said.
Mrs Foster said that on the 19th of December she had apologised to the people of Northern Ireland for her, "responsibility as minister for the renewable heat incentive" adding, "and I do so again today."
She said she had made a full statement to Stormont "outlining all aspects of the scheme and answering questions" but said, "Sinn Féin and the other parties didn't show up."
She said Sinn Fein had blocked her efforts to hold an immediate inquiry into the scheme and claimed, "it is clear that Sinn Féin’s actions are not principled, they are political."
"We put forward a proposal that would have ended the cost to the Northern Ireland budget caused by the scheme at minimum cost to tax payers but Sinn Féin rejected it," she said.
Assembly members from opposition parties walked out of Mrs Foster’s 19th of December statement on the scheme because, under Stormont rules, the DUP leader had no authority to make the statement without the backing of Mr McGuinness as deputy first minister.
Mr McGuinness said he could not back her statement as her plans for investigation into the scheme did not go far enough.
Following “clear the air” meetings between Sinn Féin and the DUP last Thursday the Northern Finance Minister Máirtín Ó Muilleoir called the DUP plans for minimising the cost of the scandal to taxpayers a “sticking plaster” solution that would not work in the long term.
"Rather than seek to resolve this issue Sinn Fein would rather take the people of Northern Ireland through the uncertainty of yet another election less than 12 months after the last one," said Mrs Foster.
"Let me make it clear the DUP will always defend unionism and stand up for what is best for Northern Ireland and it appears from the Deputy First Minister’s resignation letter that is what annoys Sinn Féin the most."
Northern Ireland Secretary of State, James Brokenshire, said that unless Sinn Féin nominates a replacement in the next seven days, he would have to call an assembly election.
He said: "I would urge Northern Ireland's political leaders to take the necessary steps to work together to find a way forward and I will work with all parties and the Irish Government to this end."
Mrs Foster has previously said she will not be stepping down, and described calls for her to do so as "misogynistic".
The Democratic Unionist leader also said she was ready for an election if Sinn Féin pulled out of the power-sharing government.
Foreign Affairs Minister Charlie Flanagan says: "I regret the circumstances which have led to the decision of Deputy First Minister Martin McGuinness to resign his office.
"The substance of the RHI controversy is very much a matter for the devolved Executive and Assembly. However, the Government is very mindful of the need to protect the integrity of the principles and institutions of the Good Friday Agreement.
"In this regard, both the effective functioning of those institutions and respecting the principles of partnership and equality are of critical importance.
“If, as appears likely, new elections to the assembly will now be required, it behoves all parties to act responsibly in word and deed, so that the political institutions of the Agreement will not be damaged in the longer term."
Mr Flanagan says he has spoken to Mr McGuinness and the Secretary of State for Northern Ireland, James Brokenshire.
Mr Flanagan continued: "As a co-guarantor of the Good Friday Agreement, the Irish Government will continue to work with the British Government and the political parties to advance political stability, reconciliation and economic prosperity in Northern Ireland."
In his letter of resignation, Mr McGuinness says: "The minister responsible for the RHO scheme should have no executive role in overseeing how this will be rectified".
He also says there are "significant conflict of interest issues", and that he urged Mrs Foster to stand aside without prejudice.
The letter also states that the Sinn Féin party will not nominate to fill the position, adding that an election is now needed "to allow the people to make their own judgement on these issues".
An election must be called in seven days if Sinn Féin do not nominate a replacement for Mr McGuinness.
The announcement by Mr McGuinness also comes just days after Sinn Féin leader Gerry Adams indicated his party was preparing to pull out of the power-sharing Stormont Executive.
Speaking in Belfast on Saturday, Mr Adams said: "If the First Minister does not take the actions that society desires and deserves and which a sustainable process of change requires, then Sinn Féin will bring this ongoing and totally unacceptable state of affairs to an end."
In a statement this evening Mr Adams said Mr McGuinness had taken "decisive action" as a result of DUP's "handling of the RHI scandal and attitude to power sharing."
"In spite of the provocation, disrespect, and arrogance from the DUP and the failures of the British government to fulfil its responsibilities over that time, Martin McGuinness has always put the people and the political process first," he said.
“This is in contrast to the DUP who have been acting to undermine equality and partnership.
“The money squandered in the RHI project belonged to unionists as well as other taxpayers. It is money which should have been used to end poverty and disadvantage or to build public services. No minister responsible for such bad governance in any other administration would be still in office."