President Donald Trump is facing calls for a show of strength toward North Korea after Pyongyang’s weekend launch of a ballistic missile, posing the first major challenge to his administration by a foreign leader and an awkward balancing act with China.

U.S. lawmakers called for military exercises with regional allies, a rapid deployment of regional missile defenses and tough new sanctions. A senior administration official said the White House is unlikely to use authority it has to go after Chinese companies that do business...