In 2008, Russia and Georgia fought a brief war over the region, which Putin visited on Tuesday, the anniversary of the start of the fighting.
"The United States urges Russia to withdraw its forces to pre-war positions per the 2008 ceasefire agreement and reverse its recognition of the Georgian regions of Abkhazia and South Ossetia," the State Department said yesterday, referring to another separatist region.
Abkhazia is internationally recognized as part of ex- Soviet Georgia, but Russia sees it as a separate country -- along with South Ossetia -- following the war with Tbilisi.
During his visit to the Black Sea resort town of Pitsunda, Putin stressed that Russia "firmly guarantees the security and self-sufficiency of Abkhazia, its independence. I am sure that this will continue in the future."
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Georgia reacted furiously, calling his visit a "cynical action" and saying it represented a continuation of Moscow's "deliberate policy against Georgia."
Yesterday, the United States reiterated its full support for Georgia's "sovereignty and territorial integrity within its internationally recognized borders."
Ties between Washington and Moscow are at a low point, largely over claims that Russia interfered in the 2016 US presidential election, in a bid to tip the contest in Donald Trump's favor, but also over Russia's role in Ukraine and Georgia.