Mitt Romney has said Barack Obama is trying to "ingratiate himself with the Kremlin" after the Russian president suggested the Republican presidential front-runner was living in a bygone era.
"The Russians clearly prefer to do business with the current incumbent of the White House," Mr Romney wrote in an opinion piece on the website of the magazine Foreign Policy.
The article represented an escalation of his attacks on Mr Obama after the US president's private remark to Mr Medvedev was caught on tape.
Mr Obama told the outgoing Russian president that he needed "space" to deal with missile defence issues because he would have more "flexibility" after the November elections.
Mr Romney drew criticism from Mr Medvedev and House Speaker John Boehner, the country's top-ranking Republican, after he said in a CNN interview that Russia was America's "No 1 geopolitical enemy".
"While the president is overseas, I think it's appropriate that people not be critical of him or of our country," Mr Boehner told reporters when he was asked if he agreed with Mr Romney.
"Clearly what's going on in Russia over the last couple of years raises some concerns."
Mr Medvedev said Mr Romney's comments on CNN "smacked of Hollywood".
He advised the White House hopefuls, including Mr Romney, to "rely on reason, use their heads", adding: "That's not harmful for a presidential candidate."
"It's 2012, not the mid-1970s, and whatever party he belongs to, he must take the existing realities into account," Mr Medvedev said.
In the opinion piece, Mr Romney expanded his criticism of Mr Obama, calling his entire foreign policy "a sad replay of Jimmy Carter's bungling" and claiming that Mr Obama had "demonstrated breathtaking weakness".
He also criticised Mr Obama for calling Russian president-elect Vladimir Putin to congratulate him on winning the presidential election earlier this year.
"It is not an accident that Medvedev is now busy attacking me," Mr Romney wrote.
Democrats have been aggressively defending the president since his off-mic comment became public, citing foreign policy experts who have called Mr Romney's comments potentially dangerous or reckless.
"The level of naivete about foreign relations that Governor Romney displays is astounding. Worse, it is potentially dangerous for our country," said Timothy Roemer, a former ambassador to India who served on the September 11 Commission.