SEOUL—North Korea’s foreign minister said the country could detonate a hydrogen bomb over the Pacific Ocean in response to President Donald Trump’s speech before the United Nations that warned the U.S. would annihilate North Korea if forced to defend itself or its allies.

The threat, made in remarks by North Korean Foreign Minister Ri Yong Ho in New York, would mark a dramatic escalation in action from Pyongyang, which in the past month has already launched two intermediate-range ballistic missiles over Japan and tested what...