Acting U.S. Secretary of State for East Asia and the Pacific Susan Thornton slammed the whole notion of a North Korean hydrogen bomb test as "outrageous" and warned it would spark a "concerted and determined global response".
North Korea's foreign minister later said that could be a hydrogen bomb test in the Pacific Ocean if Kim orders it.
The September test was North Korea's sixth and most powerful detonation, triggering a much stronger 6.3-magnitude quake that was felt across the border in China.
His fiery address to the UN General Assembly drew an unprecedented response from the despotic Kim, who delivered a personal address to his arch nemesis.
"The solution must be political".
During Tuesday's speech, Trump said there will be no choice but to "totally destroy" North Korea if it attacks, and Tillerson said on Friday that the president "takes the responsibility of the security of the American people very seriously".
In a speech before the United Nations earlier this week, Trump called Kim a "Rocket Man on a suicide mission", and threatened to "totally destroy" the North if the U.S.is forced to take military action.
Trump's executive order expanded the Treasury Department's ability to target anyone conducting significant trade in goods, services or technology with North Korea, and to ban them from interacting with the US financial system. Shortly after Kim made his statement, North Korean Minister of Foreign Affairs Ri Yong Ho said his country may consider testing a hydrogen bomb in the Pacific Ocean, according to South Korea's Yonhap News Agency.
Overnight, seismic activity was detected near the area where North Korea conducted its most recent nuclear test, and China's state news agency said it was because of an explosion.