The State and Defense departments provided backup Wednesday to President Trump’s threat a day earlier to rain down “fire and fury like the world has never seen” if North Korea did not curb its nuclear programs, but there was little sign Pyongyang was seeking to ease its threats against the U.S. and its allies in the region.

Secretary of State Rex W. Tillerson and Secretary of Defense James Mattis offered stern words for the North in the wake of reports that the regime of Kim Jong-un may have developed a nuclear device small enough to fit on a missile that could reach much of the U.S. homeland.

“What the president is doing is sending a strong message to North Korea in language that Kim Jong-un can understand,” Mr. Tillerson said. “The president just wanted to be clear to the North Korean regime.”

Mr. Mattis offered his own blunt message by urging North Korea to “stand down its pursuit of nuclear weapons” or face a U.S. response “that would lead to the end of its regime and the destruction of its people.”

North Korea’s initial response was to say it was updating plans targeting the “waters around Guam,” where the U.S. maintains a major military base. North Korean authorities also organized a large rally in Pyongyang on Wednesday as a show of defiance against tough economic sanctions approved Saturday by the U.N. Security Council.

Gen. Kim Rak-gyom, who heads North Korea’s rocket command, told state media early Thursday that Mr. Trump’s threat was “a load of nonsense” and that only absolute force could deter the American president.

Despite the biting rhetoric and criticism, however, there was a palpable lessening of the fears that a military clash on the Korean Peninsula was imminent. The response in South Korea and Japan, U.S. allies already within range of the North’s nuclear and conventional arsenal, was noticeably more muted.

There was also little sign of panic on the island of Guam, where the beaches and tourist hotels were reportedly full on Wednesday. Guam Gov. Eddie Baza Calvo recorded a YouTube video designed to try to allay any concerns residents might have, saying, “I want to reassure the people of Guam that currently there is no threat to our island or the Marianas.”

Mr. Tillerson said the president was justified because North Korea’s threatening rhetoric had “ratcheted up louder and louder” in response to growing international pressure over its nuclear and missile programs. He also noted he was passing through Guam on his way home from a regional summit in the Philippines and never considered altering his route because of Pyongyang’s threat.

“I do not believe that there is any imminent threat, in my own view,” Mr. Tillerson told reporters.

In a direct warning to Pyongyang not to escalate the situation, Mr. Mattis said the North Korean regime “would lose any arms race or conflict it initiates.”

The White House insisted that the president and his top security and diplomatic advisers were on the same page as the confrontation flared up. Spokeswoman Sarah Huckabee Sanders said Mr. Trump’s advisers were not surprised by the sharpness of his tone Tuesday.

“The words were [Mr. Trump‘s] own,” Mrs. Sanders said in a statement. “The tone and strength of the message were discussed beforehand.”

Lowering the tension

North Korea has conducted 14 missile tests this year, including two test-firings of intercontinental ballistic missiles last month potentially capable of reaching not just Guam but also the U.S. mainland.

But some analysts in Washington — including some foreign policy hawks — sought Wednesday to lower the hype and media hysteria surrounding the U.S.-North Korea tensions.

“There is no question that North Korea poses a major threat to its neighbors and can drag the United States and potentially China into a serious regional conflict,” longtime national security analyst Anthony H. Cordesman said in an assessment for the Center for Strategic and International Studies. “At the same time, no one should exaggerate the threat to the point of panic or make North Korea into some kind of towering threat.”

U.S. allies Japan and South Korea also appeared eager to downplay the threats from the Trump administration. South Korea’s Yonhap News Agency said the government in Seoul would continue to push for a peace deal with Pyongyang and maintained that the South Korean president’s office did not believe a crisis was imminent.

In Japan, government spokesman Yoshihide Suga spent more time answering questions on Wednesday about a dispute with the U.S. over the safety of its Osprey military aircraft than about North Korea, according to a report by Bloomberg News, which cited a senior Japanese official as saying few people in the government in Tokyo were taking Mr. Trump’s comments seriously.

Mr. Tillerson told reporters that mounting international pressure, including from China and Russia, will work to persuade the North Koreans “to reconsider the current pathway they’re on and think about engaging in a dialogue.”

“Americans should sleep well at night [and] have no concerns about this particular rhetoric of the last few days,” he said.

But the tough talk from Mr. Trump and his advisers has made Democrats nervous. Mark R. Warner, Virginia Democrat and vice chairman of the Senate Select Committee on Intelligence, urged the president to “think through all of our options.”

“Improvising our way into a shooting war on the Korean Peninsula without a plan puts us all at risk,” Mr. Warner said.

Mr. Trump and Mr. Mattis said Wednesday that upgrading the U.S. nuclear arsenal was a priority even before Mr. Trump took office and was made all the more urgent by the growing threat from North Korea.

“My first order as president was to renovate and modernize our nuclear arsenal. It is now far stronger and more powerful than ever before,” Mr. Trump said on Twitter. “Hopefully we will never have to use this power, but there will never be a time that we are not the most powerful nation in the world!”

The U.S. has embarked on a $1 trillion 30-year plan that was put into motion under President Obama to upgrade its nuclear warheads and delivery systems, including new ballistic missile submarines, land-based missiles and long-range stealth bombers.

The size and capabilities of North Korea’s nuclear arsenal aren’t known, although it’s believed that Pyongyang has enough weapons-grade nuclear material to build at least 10 bombs. North Korea says it has successfully tested five nuclear devices, and the DIA analysis completed late last month suggested that the North has far more nuclear devices than previously thought.

Mr. Mattis said Mr. Kim “should take heed of the U.N. Security Council’s unified voice, and statements from governments the world over [that] agree [North Korea] poses a threat to global security and stability.”

Some financial analysts also blamed the rising war of words between Washington and Pyongyang for a second day of losses in stock markets Wednesday after a string of record highs. The Dow Jones industrial average fell 36 points to close at 22,048 Wednesday.

Mr. Trump’s threatening rhetoric on North Korea “is almost entirely responsible for the pullback,” said Randy Frederick, vice president of trading and derivatives for Charles Schwab.

• Carlo Muñoz and S.A. Miller contributed to this article, which is based in part on wire service reports.

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