"We will never surrender America's sovereignty to an unaccountable, unelected global bureaucracy", Trump vowed.
"America is-so true", he said, smiling and pausing for more laughter from the General Assembly.
Every solution to the humanitarian crisis in Syria must also include a strategy to address the brutal regime that has fueled and financed it: the corrupt dictatorship in Iran.
"Iran's leaders sow chaos, death and destruction".
Trump says Iran does not respect its neighbours or their borders. "Instead, Iran's leaders plunder the nation's resources to enrich themselves and spread mayhem across the Middle East and far beyond".
Mr Trump also boasted of USA economic and military might, a rejection of "global governance". That outreach led to a historic meeting between Trump and North Korean leader Kim Jong Un in June; at the United Nations on Tuesday, Trump said there have been "encouraging" signs toward denuclearization.
Hours before taking to the podium, Mr Trump tweeted that he did not have any plans to meet with Mr Rouhani, but says he remains open to the idea. "And it's why we believe - and of course, I'm speaking in secular terms here - that the Constitution is the highest authority that we recognize". The "America first" remarks drew on a similar speech he delivered at last year's United Nations, but included more detailed examples of how that vision informs his policies on trade, immigration and the world's hot spots.
The US President is due to deliver a speech in NY on Tuesday morning, during which he is likely to touch on the Iran nuclear deal.
Trump didn't fair much better with the phrase "human smuggling" as he delved into the hardline anti-immigration portion of his overly nationalistic address.
Mr Trump used his speech to call for worldwide trade reforms and insist that his main objective as president is to protect American sovereignty.
Regarding Iran, the president indicated he won't be meeting with the Islamic Republic's leader, President Hassan Rouhani, until his nation renounced its "bloody agenda".
"We, the people, are sovereign in America, so that infringements on our sovereignty are not infringements on abstractions or infringements on the government".
The accord, negotiated under Democratic U.S. President Barack Obama, lifted most worldwide sanctions against Tehran in exchange for Iran curbing its nuclear program.
With the Iranian economy showing increasing signs of strain, the Trump administration is hoping that Tehran will have no option if it wants to survive and return to the negotiating table.
"The United States is committed to a future peace and stability in the region, including peace between the Israelis and Palestinians", he said.
Moon and Abe met on the sidelines of the United Nations assembly and the South Korean leader highlighted the importance of improved relations between Tokyo and Pyongyang to accelerate North Korean denuclearization, South Korea's Yonhap news agency reported.
Rouhani is also under increasing pressure from Iranian hardliners, including Iran's elite Revolutionary Guards, which have kept up the anti-American rhetoric ahead of the United Nations session.