Cheney flew to Kiev from Georgia, where he denounced Russia's "illegitimate, unilateral attempt" to redraw the U.S. ally's borders by force.
"Georgia will be in our alliance," Cheney told reporters while standing alongside Georgian President Mikhail Saakashvili, whose pro-Western government has sought to join NATO despite Russian opposition.
Angry Russian officials have repeatedly said U.S. military aid was instrumental in emboldening Georgia to try to retake South Ossetia by force on Aug. 7. The attack sparked five days of fighting and resulted in Russian forces driving into South Ossetia and on into Georgia.
Speaking in Moscow, Konstantin Kosachyov, head of foreign affairs committee in the Kremlin-controlled lower house, accused Cheney of trying to forge an "anti-Russian axis."
"It's Cheney who was behind all recent events on the former Soviet turf," Kosachyov said in televised remarks.
The vice president's trip was intended as a signal that the U.S. will continue cultivating close ties with Georgia and its neighbors even after Russia showed it was willing to use military force against countries along its border.
Before Georgia, Cheney also stopped in oil-rich Azerbaijan.
There are concerns the Kremlin might next seek to squeeze Ukraine as it tries to reclaim dominance in the former Soviet Union. The strategically located country of 46 million has pipelines that carry Russian gas to European consumers and a Black Sea port that is home to a key Russian naval base.
"America will do its duty to work with the governments of Georgia and our other friends and allies to protect our common interests and to uphold our values," Cheney said.
"Russia's actions have cast grave doubts on Russia's intentions and on its reliability as an international partner," Cheney added.
On the eve of his arrival, the White House announced a $1 billion commitment to help the small, strategically located nation recover from its war with Russia.
Saakashvili said Georgia was grateful for the aid, which matched his government's estimate of war damages: "Together with our other partners, in Europe, America and elsewhere, we will rebuild Georgia. We feel that we are not alone."
The United States is at Georgia's side, Cheney said, "as you work to overcome an invasion of your sovereign territory and an illegitimate, unilateral attempt to change your country's borders by force, that has been universally condemned by the free world."
New U.S. military aid to Georgia would further test relations between Washington and Moscow, which are already at a post-Cold War low.
Russia has condemned the U.S. use of warships to deliver aid as a form of gunboat diplomacy. The flagship of the U.S. 6th Fleet in the Mediterranean, the USS Mount Whitney, sailed into the Black Sea on Wednesday with more aid for Georgia.
According to a military official, the ship is planning to dock in the Black Sea port of Poti. The official, who spoke Thursday on condition of anonymity because of the sensitivity of the issue, said that the Russians have said they would not impede the ship's movement, but they want to check the cargo when it arrives in Poti. The U.S. has agreed to that, the official said.
The United States and European Union say Russia has failed to meet its obligations under an EU-brokered cease-fire deal. But Moscow insists the cease-fire accord lets it run checkpoints in security zones more than 4 miles into Georgian territory.
Italian Foreign Minister Franco Frattini said before meeting with his Russian counterpart Thursday that the EU hopes the Russian troops will pull out by Monday when an EU delegation led by French President Nicolas Sarkozy visits Moscow.
"The EU peace deal talks about temporary security measures and patrols, not about permanent installations," Frattini was quoted as saying by the ANSA news agency.
But Russia's Foreign Minister Sergey Lavrov firmly said after talks with Frattini in Moscow that Russian peacekeepers will stay in the security zone until there is a comprehensive mechanism of international monitoring.
For the first time since the outbreak of hostilities, Russia on Thursday allowed military monitors from the Organization for Security and Cooperation in Europe to patrol a road near South Ossetia. "Access to the area has been a focus for the OSCE and international partners, and we welcome this important step," Ambassador Terhi Hakala, the head of the OSCE mission to Georgia, said in a statement.
The OSCE has decided to increase its team of observers in Georgia from 8 before the conflict to 100.
Ukraine's President Viktor Yushchenko, who has supported Georgia, has objected to Russia using its ships stationed in the Ukrainian base in the war, thus dragging Ukraine into the conflict. His move has angered Moscow and further strained relations which already have been tense over energy disputes and the Russian navy presence in Ukraine.
Since the war in Georgia last month, Russia has asserted it has what President Dmitry Medvedev called "privileged interests" in its sphere of influence, which includes the former Soviet states in the South Caucasus.
"I would like to say firmly: we are worried about the Russian president's recent use of the term 'regions of privileged interests of the Russian Federation'," Yushchenko said Thursday at a meeting with the ambassadors of G-7 nations. "I don't think this corresponds to the spirit of our neighborly relations."
Cheney's visit comes at an awkward time for Yushchenko. The governing coalition, made up of his party and that of his 2004 Orange Revolution partner - now Prime Minister Yulia Tymoshenko - has collapsed, dashing hopes for quick progress and integration with the European Union.
Yushchenko and Tymoshenko have turned into bitter rivals before the 2010 presidential election, in which they are likely to compete against each other, blocking each other's policies and stalling much-needed reform.
After Tymoshenko sided with the Russia-friendly opposition to trim presidential powers, Yushchenko's allies pulled out of the coalition, robbing it of a parliamentary majority. The parliament now has to come up with a new alliance or a new election will be called. That would be the third parliamentary vote in as many years and a major embarrassment to Yushchenko's government.