Sen. John McCain said Tuesday that the Obama administration has failed to understand that Russian President Vladimir Putin views his relationship with the U.S. through a Cold War prism, urging President Obama not to back down from Russia's recent aggression in Ukraine's Crimea region.

Putin "believes that even though the president and [Secretary of State] John Kerry believe this is not [an] East-West [conflict], that this is not the Cold War, that's exactly what Putin has treated it as and why there's been a fundamental misreading of Vladimir Putin, his intentions and the things that he will do," the Arizona Republican said on MSNBC's "Morning Joe."

"There's no doubt that [Putin] will not give up in Crimea."

McCain urged Kerry, who on Tuesday arrived in the Ukrainian capital of Kiev armed with the offer of a $1 billion loan to help shore up the country's fledging government, to press the message that the U.S. "will do everything in our power to try to prevent now Putin from encroaching on eastern Ukraine."

The Republican said the White House must amp up efforts to counter Russian influence in Eastern Europe, such as pushing for the small Caucasus nation of Georgia — formerly part of the Soviet Union — to be a full member of NATO.

He also suggested that restarting an anti-ballistic missile defense program with Poland and the Czech Republic that the administration canceled, as well as possibly holding NATO military exercises in the Baltic countries of Latvia, Estonia and Lithuania — also former Soviet areas — would do much to warm Putin that "the excuse of moving into countries because of Russian populations there is a violation of every international law."

McCain also said former President George W. Bush's reluctance to intervene when Russia invaded neighboring Georgia in 2008 emboldened Putin to send troops last week into Crimea, where Russian warships are based in the Ukrainian port city of Sevastopol.

I "felt very strongly about that we didn't do enough [during the invasion of Georgia]. I think that's very clear," he said. "Vladimir Putin will never give up Sevastopol."