UNITED NATIONS -- North Korea's top diplomat said Monday that President Trump's tweet that leader Kim Jong Un "won't be around much longer" was "a declaration of war" against his country by the United States.

"This is clearly a declaration of war," Foreign Minister Ri Yong Ho told reporters through a translator in New York. "... The U.N. Charter stipulates individual member states' rights to self-defense. Since the United States declared war on our country, we will have every right to make countermeasures, including the right to shoot down the United States' strategic bombers even when they're not yet inside the airspace border of our country."

Later, at the White House, press secretary Sarah Huckabee Sanders told reporters that Mr. Trump's remarks should not be viewed as a declaration of war.

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"We've not declared war on North Korea, and frankly the suggestion of that is absurd," Sanders said.

Sanders also responded to Ri's threat to shoot down American bombers.

"It's never appropriate for a country to shoot down another country's aircraft when it's over international waters," Sanders said. "Our goal is still the same: We continue to seek the peaceful denuclearization of the Korean Peninsula. That's our focus, doing that through both the most maximum economic and diplomatic pressures as possible at this point."

Mr. Trump didn't respond to questions from reporters about Ri's comments following an Oval Office ceremony where the president signed a presidential memorandum on funding for science, technology, engineering and math education.

CBS News foreign affairs analyst Pamela Falk reports the North Korean issue was the subject of several talks at the annual U.N. General Assembly in New York, which concluded Monday. Mr. Trump met with the president of South Korea and the prime minister of Japan about North Korea last week.

"The real question is Russia and China," Falk said on CBSN. "Their foreign ministers at this U.N. General Assembly made clear what they want is more diplomatic effort."

Monday was not the first time North Korea has spoken about a declaration of war between the two countries. In July 2016, Pyongyang said U.S. sanctions imposed on Kim were "a declaration of war."

Ri referred Monday to Mr. Trump's tweet Saturday that said: "Just heard Foreign Minister of North Korea speak at U.N. If he echoes thoughts of Little Rocket Man, they won't be around much longer!"

Just heard Foreign Minister of North Korea speak at U.N. If he echoes thoughts of Little Rocket Man, they won't be around much longer! — Donald J. Trump (@realDonaldTrump) September 24, 2017

Mr. Trump also used the derisive "Rocket Man" reference to Kim in his speech at the U.N. on Sept. 19, but this time he added the word "little."

The foreign minister's brief statement to a throng of reporters outside his hotel before heading off in a motorcade, reportedly to return home, built on the escalating rhetoric between Kim and Mr. Trump.

"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," Mr. Trump had told world leaders. "Rocket Man is on a suicide mission for himself and for his regime."

Kim responded with the first-ever direct statement from a North Korean leader against a U.S. president, lobbing a string of insults at Mr. Trump and calling him a "mentally deranged U.S. dotard," a word to describe an old person who is weak-minded.

Mr. Trump responded by tweeting that Kim is "obviously a madman who doesn't mind starving or killing his people."

Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before! — Donald J. Trump (@realDonaldTrump) September 22, 2017

Kim retorted that Mr. Trump would "pay dearly" for his threat to destroy North Korea and said his country will consider the "highest level of hard-line countermeasures in history."

Asked about countermeasures, Ri then told reporters in New York that "I think it could be the most powerful detonation of an H-bomb in the Pacific."

In his speech Saturday to the General Assembly, Ri said Mr. Trump's "rocket man" insult makes "our rocket's visit to the entire U.S. mainland inevitable all the more."

"None other than Trump himself is on a suicide mission," Ri had said. "In case innocent lives of the U.S. are lost because of this suicide attack, Trump will be held totally responsible."

On Monday, Ri escalated the threat.

He opened his remarks to reporters in Korean by saying that over the last few days, the U.N. and the international community clearly have wished "that the war of words between the DPRK and the United States will not turn into real action."

DPRK refers to the North's official name, the Democratic People's Republic of Korea.

"However, that weekend, Trump claimed that our leadership wouldn't be around much longer, and ... he declared the war on our country," Ri said.

"Given the fact that this comes from someone who is currently holding the seat of (the) United States presidency, this is clearly a declaration of war," the foreign minister said.

A man watches a television screen showing President Trump and North Korean leader Kim Jong Un during a news program at the Seoul Train Station in Seoul, South Korea, on Thu., Aug. 10, 2017. AP

He said all U.N. members and the world "should clearly remember that it was the U.S. who first declared war on our country."

Ri then said North Korea now has the right to retaliate against U.S. bombers.

He ended his brief remarks by saying: "The question of who won't be around much longer will be answered then."

Mr. Trump's tweets have sparked or stoked several controversies during the first year of his presidency, including his recent criticism of NFL players who kneel during the national anthem as a form of protest.

Even some of the president's supporters aren't fans of how he uses Twitter, as one told "60 Minutes" special contributing correspondent Oprah Winfrey in a roundtable discussion broadcast Sunday night.

"I still don't like his attacks, his Twitter attacks, if you will, on other politicians," a man named Tom said. "I don't think that's appropriate. But, at the same time, his actions speak louder than words. And I love what he's doing to this country. Love it."

Meanwhile, North Korea was trying to convince other governments to condemn Mr. Trump for his U.N. speech, CBS News correspondent Ben Tracy reports.

In a letter sent to foreign parliaments, North Korea called Mr. Trump's threat an "intolerable insult," North Korean state media reported, and Ri said Mr. Trump's words made North Korea's "rockets' visit to the U.S. mainland inevitable all the more."

It was not immediately clear which governments had been sent the letter, Tracy reports, but it was part of what appeared to be a new approach of trying to turn Mr. Trump's threats to destroy North Korea against him.

Also making headlines Monday is North Korea's foreign minister saying that the derisive words Mr. Trump has been using against dictator Kim Jong Un amount to a declaration of war, and the North has the right to retaliate by shooting down U.S. bombers, even in international air space, CBS News' David Martin reports.

Over the weekend, U.S. military planes flew up the coast of North Korea in a show of force. The B-1 bombers and their F-15 fighter escorts flew at night, when any North Korean jet that attempted to intercept them would be at an extreme disadvantage.

And although they flew further north than American combat aircraft have gone in years, they stayed 200 miles off the coast -- out of effective range of North Korean anti-aircraft missiles. In other words, there was very little, if any, risk North Korea could actually shoot down an American bomber.