In their final debate before the Nov. 6 presidential election, Romney blamed Obama for what he said has been "a pretty drastic reversal of our hopes" for the Middle East since the outbreak of the Arab Spring uprisings.
The Republican nominee said while "we can't kill our way out of this mess," if elected he would "go after the bad guys to make sure we do our very best to take them out of the picture."
He said the policy the United States must pursue "is to get the Muslim world to reject extremism on their own" to reverse "the rising tide of chaos."
Romney said the way to do that is to help them build their economies, improve education, encourage religious and gender equality, and promote the rule of law.
Obama panned Romney's worldview, chiding him for saying Russia is the biggest threat to America.
"The 1980s are calling and are asking for their foreign policy back," Obama said, a reference to the Cold War being over for decades.
"Governor, I know you haven't been in the position to actually execute foreign policy, but every time you've offered an opinion, you've been wrong," Obama said, citing Romney's position on Iraq, Russia and Afghanistan.
Obama said Romney's changing positions send "mixed messages to both our troops and our allies." He said America needs "strong, steady leadership, not wrong and reckless leadership that's all over the map."
"That's not a recipe for American strength," Obama said.
Romney later said the greatest international threat to the United States is a nuclear Iran.
The Democratic president cited his ending of the war in Iraq and the transition of U.S. forces out of Afghanistan.
"We are stronger now than when I came into office," Obama said. "Our alliances have never been stronger."
On Syria, Romney said he would provide effective leadership that would identify responsible parties among the rebel forces seeking to topple Syrian President Bashar Assad, help them form a guiding council and make sure they get the weapons they need to fight on.
"I don't want our military involved in Syria," Romney said, adding he doesn't think that will be necessary.
He faulted Obama for working through the United Nations and Russia in the past year instead of taking a lead role.
Obama said he has been careful to make sure arms only go to opposition forces that ultimately will be friendly to the United States.
Both men said they stand firmly against Iran getting a nuclear weapon capability and will stand with Israel, the United States' closest ally in the Middle East.
Romney said he would not cut any military spending and lamented the small size and aging equipment of the U.S. Navy and Air Force, which he said are at 1917 levels. He said he would not allow the $1 trillion in cuts called for under congressional legislation to happen, something Obama also said won't happen.
Obama said unfocused spending on the military isn't the best way to protect America, saying the number of ships isn't the most accurate way to measure military strength.
"We also have fewer horses and bayonets because the nature of our needs has changed," Obama said.
On China, the president said he sees both an adversary to a potential partner, "if it's following the rules."
"So my attitude coming into office was that we are going to insist that China plays by the same rules as everybody else," the president said.
"I know Americans had seen jobs being shipped overseas; businesses and workers not getting a level playing field when it came to trade. And that's the reason why I set up a trade task force to go after cheaters when it came to international trade. That's the reason why we have brought more cases against China for violating trade rules than the other -- the previous administration had done in two terms. And we've won just about every case that we've filed, that has been decided."
Romney said China is like the United States in that it wants a stable world that is "free and open."
"They don't want war. They don't want to see protectionism. They don't want to see the world break out into ... into various forms of chaos," Romney said.
But, he said, China has not played by the rules.
"I've watched year in and year out as companies have shut down and people have lost their jobs because China has not played by the same rules, in part by holding down artificially the value of their currency," the Republican nominee said. "It holds down the prices of their goods. It means our goods aren't as competitive and we lose jobs. That's got to end.
"That's why on Day 1, I will label them a currency manipulator, which allows us to apply tariffs where they're taking jobs. They're stealing our intellectual property, our patents, our designs, our technology, hacking into our computers, counterfeiting our goods.
"They have to understand we want to trade with them. We want a world that's stable. We like free enterprise, but you got to play by the rules.
"I want a great relationship with China. China can be our partner, but -- but that doesn't mean they can just roll all over us and steal our jobs on an unfair basis."
Obama countered that Romney, as a businessman, was responsible for shipping U.S. jobs overseas "because you invested in companies that were shipping jobs overseas."
"If we had taken your advice Governor Romney about our auto industry, we'd be buying cars from China instead of selling cars to China," Obama said.
"If we take your advice with respect to how we change our tax codes so that companies that earn profits overseas don't pay U.S. taxes compared to companies here that are paying taxes. Now that's estimated to create 800,000 jobs, the problem is they won't be here, they'll be in places like China.
"And if we're not making investments in education and basic research, which is not something that the private sector is doing at a sufficient pace right now and has never done, then we will lose the (inaudible) in things like clean energy technology.
"Now with respect to what we've done with China already, U.S. exports have doubled since I came into office, to China and actually currencies are at their most advantageous point for U.S. exporters since 1993."
The debate in Boca Raton, Fla., faced some stiff sports competition: football and baseball. The debate at Lynn University went up against ESPN's Monday Night Football game featuring the NFC North Division rivals Detroit Lions and Chicago Bears, as well as the decisive Game 7 between the St. Louis Cardinals and the San Francisco Giants in baseball's National League Championship Series.
The competing sports events threatened to siphon off ratings for the debate on foreign policy, USA Today reported.
The first Obama-Romney debate Oct. 3 drew 67.2 million views, Nielsen ratings indicated, while the second faceoff had 65.6 million television viewers.
The 90-minute debate was moderated by Bob Schieffer, host of CBS News' "Face the Nation."