To Mr. Obama’s critics, the root of the seeming absence of American leverage in the Middle East today is a light footprint that was simply too light.

“I think the way to understand Obama’s approach — I wouldn’t call it a strategy — is that he has a uniform preference to keep most problems at a distance,” said Eliot A. Cohen, a professor at the Johns Hopkins School of Advanced International Studies who worked for Mitt Romney’s presidential campaign and helped develop Mr. Romney’s critique of Mr. Obama’s approach. “That is what the light footprint has been all about. And it’s run out of gas.”

Libya has become Exhibit 1 in that argument. Mr. Obama reluctantly committed air power to the ouster of Col. Muammar el-Qaddafi over the objections of the secretary of defense at the time, Robert M. Gates, who warned that there was no direct American interest in the outcome. Instead, the president urged the Arab League and NATO to “put skin in the game.”

They did, but Mr. Obama’s reluctance to put American forces on the ground during the fight, and his decision to keep America’s diplomatic and C.I.A. presence minimal in post-Qaddafi Libya, may have helped lead the United States to miss signals and get caught unaware in the attack on the American mission in Benghazi. Military forces were too far from Libya’s shores during the Sept. 11 attack to intervene.