Ri, who attended this year's UN General Assembly session, said the international community had hoped that a "war of words" would "not turn into real actions".
"However, last weekend, Trump claimed our leadership would not be around much longer," Ri said. "He declared a war on our country." Later on Monday, the White House rejected Ri's interpretation of Trump's tweets.
"We have not declared war against North Korea and frankly the suggestion of that is absurd," White House spokeswoman Sarah Huckabee Sanders said.
Alarm over Pyongyang's nuclear and ballistic missile programmes dominated this year's gathering of world leaders at the UN, amid fears the heated rhetoric could accidentally trigger a war.
Those fears were sharpened after US bombers flew off the coast of North Korea on Saturday - going the furthest north of the demilitarised zone that any US aircraft has flown this century.
"Since the United States declared war on our country, we will have every right to take counter-measures including the right to shoot down US strategic bombers even when they are not yet inside the airspace border of our country," Ri said.
"The question of who won't be around much longer will be answered then." A Pentagon spokesman stressed on Monday that the bombers flew in international airspace and had every right to do so.
Fears of accidental clashAs the rhetoric heated up, South Korea appealed for an easing of tensions, with Foreign Minister Kang Kyung-wha saying that further provocations can be expected from Pyongyang but must not be allowed to get out of control. "It is imperative that we, Korea and the US together, manage the situation ... in order to prevent further escalation of tensions or any kind of accidental military clashes which can quickly go out of control," Kang said in Washington.
South Korea has reacted with unease to Trump's threat to "totally destroy" North Korea as its densely-populated capital Seoul is located just 56km from the demilitarised zone dividing the Korean Peninsula.
In his UN address last week, Trump delivered the blunt threat, deriding leader Kim Jong-un as "Rocket Man" and declaring he was "on a suicide mission".
Kim hit back with a personal attack on Trump, branding him "mentally deranged" and a "dotard" and warning he would "pay dearly" for his threat.
There have been repeated appeals for calm from the United Nations, Russia and China.
On Tuesday, China said that war on the Korean Peninsula will have no winner.
Lu Kang, the spokesman for the Chinese foreign ministry, said Beijing hopes that US and North Korean politicians can realise that resorting to military means would never be a viable way out.
Russia's foreign ministry also said on Tuesday that a military conflict on the Korean Peninsula would lead to "catastrophic consequences". It added that it would work behind-the-scenes to find a political solution to the rising tensions with North Korea, and that the US approach is a "dead in".
Earlier this month, the UN Security Council imposed new sanctions on North Korea over its sixth and most powerful nuclear test. In recent months, Pyongyang also test-fired intercontinental missiles - saying it needs to defend itself against the threat of a US invasion.
Asked about the North Korean minister's latest remarks, UN spokesman Stephane Dujarric said: "When you have the rise of tension, the rise of rhetoric, so does the risk of miscalculation." In his UN address, Ri warned that Trump's threat to destroy North Korea made "our rockets' visit to the entire US mainland all the more inevitable".
The rhetoric comes as international alarm mounts over Pyongyang's weapons ambitions - including a suggestion by Ri last week that the country is considering detonating an H-bomb over the Pacific.
US Defense Secretary Jim Mattis has said such a move would be a "shocking display of irresponsibility."