ST. PETERSBURG, Russia—President Barack Obama fell short in his bid to rally international support for a U.S. military strike on Syria, failing to overcome divisions at a summit of world leaders before flying home to face skeptics in Congress and the American public.

Mr. Obama spent two days coaxing leaders of the Group of 20 nations, pulling presidents and prime ministers aside for private conversations and making his case in a four-hour dinner at a historic palace here.

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