Medvedev said bilateral relations "are undergoing serious trials today" and emphasized the two countries' unifying cultural and religious traditions, and personal contacts.
The congress gathered to address problems facing Georgians in Russia as ties between the ex-Soviet states plunged to a new low after the armed conflict that broke out after Georgia attacked South Ossetia.
Tbilisi severed diplomatic relations with Moscow after it recognized South Ossetia and Abkhazia, the other breakaway region, as independent states after the five-day war.
Strains in relations with Russia and Georgia, run by Western-leaning leaders, have caused difficulties for hundreds of thousands of Georgian guest workers in Russia.
"Russia remains devoted to the centuries-old traditions of friendship with the Georgian people," Medvedev said.
"We sincerely want to see Georgia a stable, independent and truly democratic state. A state that would live in peace and security and maintain friendly relations with all the countries," the president said, adding he hoped the congress would contribute to improving ties between Russia and Georgia.
