DUP leader Arlene Foster should not step aside from being nominated as first minister at Sinn Féin's demand, the DUP's Simon Hamilton has said.
He said she has the "full support" of her party and the election result was an "endorsement" of her leadership.
But Sinn Féin repeated it will not form a government with the DUP if they nominate Mrs Foster as first minister before a report into RHI is published.
Mrs Foster set up the botched Renewable Heat Incentive (RHI) scheme in 2012.
The fallout over a projected £490m overspend on the green energy scheme led to the collapse of the DUP and Sinn Féin-led coalition government in January.
A snap election, held on Thursday, brought an end to the unionist majority at Stormont but the DUP remains the largest party with 28 seats.
Mr Hamilton, a former DUP economy minister, told the BBC's Sunday Politics show: "It is not for Sinn Féin to dictate who our nominee for first minister is, it is not reasonable to do so."
Unionist commentator and author, Prof Jon Tonge, told the programme the election had been an "absolute disaster for unionism".
"For the first time they've lost their overall majority in Stormont and they may never, ever regain their overall majority in Stormont.
"So, from going from a very powerful position only last May 2016, suddenly unionism is in crisis."
On Saturday, DUP MP Gavin Robinson said his party was not ruling out Sinn Féin's ultimatum but added it would be Mrs Foster's decision.
However, Mr Hamilton said Thursday's election result was an "endorsement" of Mrs Foster's leadership of the DUP and Sinn Féin should "respect" her mandate.
He said he wanted to see her lead the party into power-sharing negotiations on Monday and her to be re-appointed as first minister of a coalition government.
"I've known Arlene for many years and she's one of the most capable politicians in Northern Ireland," Mr Hamilton told the programme.
"She has my full support, she has the support of the party, and most importantly she has the support of 225,000 people across Northern Ireland who voted for the DUP, who increased the DUP's mandate."
The DUP's vote increased in an election that saw a 10% increase in the overall turnout.
However, the DUP share of the vote fell and it lost seats and a number of high-profile MLAs.
Mr Hamilton said it was "disappointing" that some of his colleagues had not been re-elected but said the DUP was still the biggest party in Northern Ireland.
Speaking on the same programme, Sinn Féin's John O'Dowd said: "We are not dictating who leads the DUP."
He added: "If the DUP decides, after the implementation talks that are going to take place over the next number of weeks, that they are going to nominate Arlene Foster as joint first minister, Sinn Féin will not support that nomination."
Mike Nesbitt is stepping down as UUP leader after a disappointing election result which saw his party overtaken by the nationalist SDLP.
Ulster Unionist Party member Robin Swann told the programme he would not rule himself out of running for the UUP leadership.
Independent Unionist, Clare Sugden, who served as Justice minister in the previous Stormont mandate, said she would be prepared to return to the ministry.