UNITED NATIONS—President Barack Obama argued the U.S. case before world leaders for resolving the Middle East's deepest conflicts, but pushback from Iran dimmed hopes that had been building for a rapid leap forward.

Iranian President Hasan Rouhani's decision against meeting Mr. Obama—or even exchanging a handshake—at the U.N. General Assembly in New York on Tuesday soured what American and European officials had hoped would mark an advancement in efforts to wind down tensions.

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