Rick Santorum, who is Romney's chief rival, said Tuesday that Obama's comments suggested he is willing to sacrifice U.S. security and the security of its allies.
"This isn't about politics. This is about the president's real agenda," Santorum said in Beaver Dam, Wis. "The president's real agenda is to withdraw, to allow — whether it's the Russians or the Chinese or whoever it is, the Iranians — let them have their run of the table because America's no longer in the business of protecting ourselves and our allies."
Republican candidate Newt Gingrich also questioned Obama's motives.
"I'm curious, how many other countries has the president promised that he'd have a lot more flexibility the morning he doesn't have to answer to the American people?" Gingrich said Monday on CNN.
Neither Obama nor Medvedev knew they were being heard when they conferred quietly at what was billed as their last meeting of Medvedev's presidency. He leaves office in May, to be replaced by the incoming Vladimir Putin.
According to ABC News, Medvedev replied in English: "I understand. I will transmit this information to Vladimir."
Obama said the way the Republicans seized on his comments only made his point that the atmosphere is too politicized right now to advance arms control with Russia.
"The only way I get this stuff done is if I'm consulting with the Pentagon, if I'm consulting with Congress, if I've got bipartisan support, and the current environment is not conducive to those kinds of thoughtful consultations," Obama said. "I think we'll do better in 2013."
There, again, Obama's remarks suggested he feels good about his re-election prospects.
AP National Security Writer Anne Gearan in Seoul, South Korea, Associated Press writers Kasie Hunt in San Diego, Philip Elliott in Beaver Dam, Wis., and Charles Babington and Donna Cassata in Washington contributed to this report.
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