The vice president told the parliament: "I know there is some concern, and I understand it, that our efforts to reset relations with Russia will come at the expense of Georgia.
"Let me be clear: they have not, they will not and they cannot."
Referring to Russia's decision to recognise the independence of the breakaway regions, Biden said: "We will not recognize Abkhazia and South Ossetia as independent states and we urge the world not to recognise them as independent states."
Prior to the war the US had given military assistance to Georgia, which included training its forces and providing various categories of arms, much of which was destroyed in the conflict.
Saakashvili has previously expressed a strong interest in once again acquiring US weapons as he seeks to rebuild Georgia's military.
But Grigory Karasin, Russia's deputy foreign minister, said in an interview with Itar-Tass, the Russian state news agency, that Moscow will not permit Georgia to re-arm.
"We are deeply concerned about the actions of the Georgian leadership aimed at the remilitarisation of the country, which are amazingly being met with a calm and even a positive reaction by some governments," Karasin said.
"We will continue to prevent the re-arming of Saakashvili's regime and are taking concrete measures against this.
David Bakradze, Georgia's parliamentary speaker, said: "The US-Georgia strategic partnership charter envisages, among other issues, that the United States will help Georgia to further develop its defence capabilities."
Eka Tkeshelashvili, Georgia's national security council secretary, said it was "wrong to say that we are discussing such a minor issue" as the supply of US arms during Biden's visit.
Tkeshelashvili said the two countries were "holding talks on a strategic level" and that Karasin's comments were "propaganda" aimed at "trying to portray Georgia as a particularly militarised country which, of course, is not true".
Saakashvili, speaking to reporters as he greeted Biden earlier, said Georgia had made an "irreversible" choice in favour of democracy and also remained committed to joining Nato.
"We are dedicated to joining Europe, the North Atlantic alliance," the Georgian president said.
Biden said the US supported Tbilisi's aspiration to join the military alliance.