The president referred to North Korean leader Kim Jong Un as "rocket man'' and threatened to "totally destroy North Korea'' if the United States is forced to defend itself or allies against the North's aggression.
"The United States has great strength and patience, but if it is forced to defend itself or its allies, we will have no choice but to totally destroy North Korea," Trump said Tuesday. "Rocket Man is on a suicide mission for himself and for his regime. The United States is ready, willing, and able, but hopefully this will not be necessary."
The UN secretary general warned UN members Tuesday morning that the threat of a nuclear attack is at its highest level since the end of the Cold War, and that fiery talk can lead to fatal misunderstandings, CBS2's Alice Gainer reported.
At the Pentagon, U.S. Secretary of Defense Jim Mattis was asked about the president's new nickname for Kim Jong Un.
"We're dealing with the North Korea situation through the international processes, and we will continue to do so with Secretary Tillerson leading the effort," he said. "We will hopefully get this resolved through diplomatic means."
Trump also slammed the Iranian government, calling it an "economically depleted rogue state'' whose chief export is violence.
Questioning the 2015 nuclear deal with Iran, Trump said the world cannot allow the "murderous regime'' to continue its destabilizing activities while building dangerous missiles. He added world leaders "cannot abide'' by the agreement if it "provides cover'' for Iran to eventually build its nuclear program.
"The Iran deal was one of the worst and most one-sided transactions the United States has ever entered into. Frankly, that deal is an embarrassment to the United States, and I don't think you've heard the last of it, believe me. It is time for the entire world to join us in demanding that Iran's government end its pursuit of death and destruction," Trump said.
French President Emmanuel Macron said that scrapping the deal would be a big mistake.
"I think that it's better than nothing," he said.
But Trump did have Israeli Prime Minister Benjamin Netanyahu on his side.
"He called it an embarrassment, while I couldn't agree with him more and here's why," Netanyahu said. "Iran vows to destroy my country. Every day."
Iran's president hasn't given any specifics about how the country would react to a withdrawal.
"Given that Mr. Trump's reactions and actions and policies are somewhat unpredictable, we have had long thought and discussions about our reactions," President Hassan Rouhani said.
Trump also said he is confident that a diplomatic dispute between Qatar and its Arab neighbors can also be solved.
"I'm sure that with your interference, hopefully we can find a solution for this problem," said Tamim bin Hamad Al Thani, emir of Qatar. "We've also said that were very open for dialogue and we will always be open for dialogue."
During his more than 40-minute inaugural address to the UN General Assembly, Trump said that he will "always put America first'' and the U.S. can no longer be taken advantage of in its dealings around the globe.
Trump said he will "defend America's interests above all else.'' He says the U.S. will "forever be a great friend to the world,'' including its allies, but the U.S. can no longer be taken advantage of and get nothing in return.
But Trump said that UN member states should unite to face global dangers, and rallied other countries to do their part in solving global issues, including sharing the burden of fighting terrorism.
"We live in a time of extraordinary opportunity... but each day also brings news of growing dangers that threaten everything we cherish and value," Trump said. "Terrorists and extremists have gathered strength and spread to every region of the planet. Rogue regimes represented in this body not only support terrorists but threaten other nations and their own people with the most destructive weapons known to humanity."
"To put it simply, we meet at a time of both immense promise and great peril," Trump said. "It is entirely up to us whether we lift the world to new heights or let it fall into a valley of disrepair."
The president talked tough on terror saying "it is time to expose and hold responsible'' nations that provide funding and safe harbor to terror groups. He says all responsible nations must work together to confront terrorists and "the Islamic extremist that inspires them.''
"We will stop radical Islamic terrorism, because we cannot allow it to tear up our nation and, indeed, to tear up the entire world," said the president.
The leaders of Russia and China, among others, were not in attendance. Trump did not mention Russia in his speech, but he will meet with Russia's foreign minister and host a reception for UN leaders Tuesday night.
After his address, Trump went to a luncheon where he gave a toast, admittedly calling himself a critic of the UN for years.
"To the potential -- the great, great potential -- of the United Nations. Thank you all for being here," he said.
In his first appearance at the United Nations on Monday, Trump made a call for sweeping reforms.
"In recent years, the United Nations has not reached its full potential because of bureaucracy and mismanagement," Trump said.
The U.S. is the UN's largest contributor, paying at least 25 percent of the operating budget. The president encouraged other nations to consider joining in, reviewing the UN's spending and efficiency.
"I think the main message is 'Make the United Nations great,' not again, 'Make the United Nations great.' Such tremendous potential and I think we'll be able to do this," Trump said.
"Major portions of the world are in conflict and some in fact are going to hell, but the powerful people in this room, under the guidance and auspices of the United Nations can solve many of these vicious and complex problems," Trump said. "The American people hope that one day soon the United Nations can be a much more accountable and effective advocate for human dignity and freedom around the world."
North Korea was a major conversation point in Monday's phone call with China's president.
Dinner with Latin American leaders was centered around trade and instability in Venezuela, which Trump addressed during his speech Tuesday.
"The Venezuelan people are starving, and their country is collapsing, their Democratic institutions are being destroyed," Trump said. "This situation is completely unacceptable and we cannot stand by and watch. As a responsible neighbor and friend, we and all others have a goal -- that goal is to help them regain their freedom, recover their country and restore their democracy."
Trump accused its President Nicolas Maduro of stealing power from elected representatives to preserve his "disastrous rule.''
There is no sign that Trump will falter on his choice to withdraw from the Paris Climate Accord unless it is renegotiated, which will likely be a major topic this week.
(© Copyright 2017 CBS Broadcasting Inc. All Rights Reserved. The Associated Press contributed to this report.)
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