President Barack Obama's 15 seconds of face time with Russian President Vladimir Putin on Thursday, while American and Russian warships patrolled the eastern Mediterranean, spoke to a deep chill that has created one of the biggest complications to the U.S.'s plan to strike Syria.

Mr. Obama used the opening day of a Group of 20 meeting in St. Petersburg to press his Syria agenda, speaking for an hour with Japan's prime minister. With Mr. Putin, the summit's host, he simply shook hands and smiled for cameras, with no further meeting plans scheduled.

Their tense standoff, in many ways, is the outgrowth of previously undisclosed calculations about the level of U.S. interest in the civil war in Syria. In early 2012, White House and State Department officials asked themselves what the U.S. might be willing to do to wean Russia from its support of Syrian President Bashar al-Assad. Curtail missile defenses in Europe? Pare plans to enlarge the North Atlantic Treaty Organization?

Their conclusion: These initiatives weren't worth sacrificing for a deal on Syria, which was then lower on the foreign-policy priority list, say current and former officials who took part in the brainstorming exercise. Likewise, officials doubted such a gambit would work with a Russian leader whose motivations have confounded the U.S.

Now, Syria is the dominant foreign-policy challenge of Mr. Obama's second term. Russia is one of the biggest complicating factors. Moscow's unwavering opposition to striking Syria is the reason Mr. Obama couldn't get United Nations Security Council backing for military action against Mr. Assad, which in turn helped prompt the administration's risky gamble to seek political and legal cover from Congress.