UNITED NATIONS—The Trump administration tried to unify world leaders behind a tougher approach toward North Korea on Friday, but risked finding itself more isolated amid stiff resistance from China and Russia, which accused the U.S. of ratcheting up tensions.

North Korea itself punctuated the debate later in the day by firing a ballistic missile, one of a series of tests that have signaled the country’s determination to move ahead with its weapons program despite international condemnation. The missile launch appeared to fail,...