MANILA—An anti-American rant by Philippine President Rodrigo Duterte, which he later recanted, cast a shadow on Washington’s bid to bolster ties with Asian allies such as Manila to help offset China’s growing might.

U.S. President Barack Obama scrapped a planned meeting with Mr. Duterte in Laos on Tuesday after the outburst, sparking strains with one of Washington’s oldest Asian allies. Mr. Duterte later expressed regret. Still, the U.S.-Philippine tensions contrasted with Mr. Duterte’s apparent determination to improve tattered...