"The United States is a stronger, safer, and richer country than it was when I assumed office less than two years ago", Trump declared.
"Iran has acted very badly", said Mr Trump.
Pyongyang has also been seeking relief from crippling US and worldwide sanctions, though Washington has said it remains too early in the process for such a move.
Tying his disparate targets together was his administration's rejection of multilateral institutions and a warm embrace of the nationalist populisms that have wracked politics around the world.
Mr Rouhani said on Monday Teheran would not talk to Mr Trump until the U.S. returned to the 2015 deal.
Activists and Democratic lawmakers were quick to react.
"Americans expect the United States to assert bold leadership on the world stage that reflects our values", he said, "and under President Trump, we are certainly leading from the front".
Trump has, in the past, warned that the country needed a president who was not "a laughing stock to the entire World" in a 2014 tweet, referring to former President Barack Obama. But Trump's assessment of the diplomatic détente with North Korea omitted the challenges that remain and the reality that North Korea continues to quietly develop nuclear weapons and has yet to take any concrete, verifiable steps toward denuclearization.
While there is overlap - particularly on the issue of chemical weapons and North Korea - Washington's European allies are raising issues that don't seem to register much on the United States priority list this year, chief among them climate change.
Bill Cosby sentenced to prison for sex assault
Cosby , 81, was found guilty on each of three counts of aggravated indecent assault of Toronto woman Constand, who is now 45. His lawyers asked for house arrest, saying Cosby - who is legally blind - is too old and vulnerable to do time in prison.
"The Iranian people are rightly outraged that their leaders have embezzled billions of dollars from Iran's treasury, seized valuable portions of the economy, and looted the people's religious endowments, all to line their own pockets and send their proxies to wage war", Trump continued. Trump reimposed strict economic sanctions to pressure Iran into talks, including measures to limit the Islamic Republic's oil exports.
"[They cannot] possess the means to deliver a nuclear warhead", he said.
Velayati also dismissed as "unlikely" the threat of Iran's oil exports dwindling to zero as a result of pressure from the USA and its allies.
Facing rising oil prices just weeks before crucial midterm elections, Trump used his United Nations speech to lambast the Organization of the Petroleum Exporting Countries, which he blamed for withholding production and jacking up crude prices.
"Virtually everywhere socialism and communism has been tried, it has led to corruption and decay", Trump said.
Maas said "There was great unanimity among those involved that we want to keep this deal alive, that we want to soften the consequences resulting from USA sanctions".
Oil prices are now above $80 a barrel-but OPEC has boosted oil output in recent months. Trump noted that since sanctions had been lifted, Iran had increased its military budget by almost 40 percent. Nothing highlights the zero-sum-mentality that is the core of everything this president says or does better than Trump's declaration that in the future USA foreign aid dollars will only flow to friendly countries.
U.S. allies in the Middle East, notably Saudi Arabia, have, however, been delighted by Trump's stance. "We will not allow our workers to be victimized, our companies to be cheated, and our wealth to be plundered and transferred", he said.
Referring to his tariff war with China, Mr. Trump said the USA has lost 3 million manufacturing jobs since China joined the WTO.
He decried "broken and bad trade deals", and then celebrated his administration's minor revisions to the existing trade pact with South Korea and shrinking America's North American free trade arrangements by more than half.
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