ISLAMABAD—Pakistan's army chief issued a rare public condemnation Thursday of a U.S. drone strike that killed up to 40 people, in a sign of mounting disagreement between the two countries over how to defeat Taliban militants sheltering on Pakistan's border with Afghanistan.

Gen. Ashfaq Kayani said an unmanned U.S. Predator drone had earlier Thursday "carelessly and callously targeted" a peaceful meeting of elders in North Waziristan, a Pakistani tribal region and Taliban stronghold.

"In...