The veteran politician was speaking during his first TV interview since sending an 11-page letter to his party withdrawing support for the Peace and Reconciliation Centre at the Maze site.
In the document, he blamed issues including Sinn Féin's involvement in an IRA commemoration in Castlederg, Co Tyrone and naming a children's play park after an IRA hunger striker for the decision, saying it had been "insensitive" towards victims.
Mr Robinson sent the letter while holidaying in Florida and, at the time, Sinn Féin representatives criticised the First Minister for making the announcement while out of the country and for not consulting with deputy First Minister Martin McGuinness.
I've made it very clear that I'm going on - that's the wish of the party.
Peter Robinson, speaking exclusively to UTV
Speaking to UTV's political correspondent Tracey Magee in New York on Monday, the DUP leader denied making a 'u-turn' over the Maze.
Mr Robinson is in the American east coast city to meet with Mr McGuinness on Monday night for the first time since his party's withdrawal of support.
They did not travel together to the conference as Mr Robinson was still in the United States on holiday.
"I've a job to do - I've made it clear that my time and exit is based on what can be achieved and when I feel it's appropriate for someone to take over," he said.
"My party had made it very clear that we didn't feel that having retained the buildings at the Maze listed was a very good idea. We opposed it - in fact, we were the only ones to do so."
Mr Robinson added that he was "prepared to run with the proposal" - providing that he continued to have a veto on the nature of the proposal.
However, he added that the Castlederg parade, which he described as glorifying terrorism, had killed it all.
"It's not a u-turn - what it is, is setting a condition and the condition is there has to be public support for it," he said.
"And I can't gain the support because it's Sinn Féin that they don't trust on the issue of what they would do with the centre."
Mr Robinson added that he would continue to do the job that he has been elected and appointed to do, before saying that he and the deputy First Minister were in New York to attract investment to Northern Ireland.
One glimmer of hope is that no matter what is said during tonight's crunch meeting - or at what volume - the very next day, both men will present a united front as they schmooze America's great and good.
UTV’s Tracey Magee, in her blog from New York
UTV's Political Editor Ken Reid said that the pair need to show unity while on the trip.
"There's a meeting tonight that's seen by some as crucial," he said.
"And why it's crucial is that they are not only sorting that out, but the fact they're on a trade mission trying to sell Northern Ireland meeting some very important business people.
"This is a hard sell job and what would it look like if the two of them have fallen out?"
He added: "They are pragmatic politicians, but don't underestimate how deep the hurt and the controversy is over that decision on the Maze.
"I would imagine at their private meeting tonight in a New York hotel that they will have some pretty straight talking with each other."