Biden said: "We do not recognise ... anyone else's right to dictate to you ... what alliance you will seek to belong to, or what bilateral relationships you have," after talks with Viktor Yushchenko, the Ukrainian president, on Tuesday.
Georgia, which also wishes to join Nato, is urging the US to help monitor its boundaries with Abkhazia and South Ossetia.
Eka Tkeshelashvili, the national security council secretary, said Georgia wants US monitors to bolster the work of the 246 European ones currently in the region.
"This would assist in Georgia's economic development, as it would signal investors that their security and safety are guaranteed and they can invest their money," she said, ahead of Biden's visit.
The EU monitors are the only remaining international ones in Georgia, but are blocked from travelling inside the breakaway regions, which Moscow has recognised as independent states.
Saakashvili's government was shaken earlier this year by mass street protests demanding his resignation, due to anger over the Russia-Georgia conflict and claims his leadership has become increasingly autocratic.
The president has said he tried to defend Georgia from Russian aggression, and announced a series of political reforms to parliament on Monday intended to address his critics' complaints.
The proposals include the direct election of mayors, a reduction in presidential powers and early local elections.
During his visit to Ukraine, Biden assured Yushchenko that US attempts to renew relations with Russia would "not come at Ukraine's expense".
Yushchenko, who came to power after a pro-Western revolution in 2004 in which he benefited from US support, told Biden that US-Ukrainian relations should be developed in a "constructive way".
He said his country had chosen "a European path", where "democracy rules", in an apparent dig at Russia.
Biden's comments came just two weeks after Barack Obama, the US president, visited Moscow in an effort to strengthen ties between the two former Cold War enemies.
Neave Barker, Al Jazeera's correspondent in Moscow, said: "The Obama administration know they are walking a political tightrope, on the one hand wanting to maintain workable relations with post-Soviet nations in Eastern Europe, but at the same time wanting to restore a new level of ties with Russia.
"I think this meeting doesn't go any further than touching base between Ukraine and the Obama administration.