MOSCOW—Russia said Monday that U.S. Secretary of State John Kerry postponed a face-to-face meeting with Russian President Vladimir Putin to discuss an American proposal on settling the dispute over Crimea, as the distance between two sides shows little sign of narrowing.

The meeting, which Russia said was supposed to happen Monday, would have been the first direct contact between the U.S. and the Russian president since Russian troops seized effective control of Crimea in late February and would have come ahead of a regionwide...