Russia's ambassador to the UN, Vitaly Churkin, has called on the Security Council to censure Georgia for "dangerous and unacceptable" actions.
Already tense relations are deteriorating further in Tbilisi
It comes amid tension over the arrest in Georgia of four Russian military officers, who are accused of spying.
Moscow has recalled its ambassador to Tbilisi and ordered the partial evacuation of Russian personnel from Georgia, which will begin later.
Georgian forces are still surrounding a Russian military building in Tbilisi.
They say another Russian military officer they want to question is sheltering inside.
A Russian contract servicemen - who was detained together with the four officers - was released on Friday, Georgian and Russian officials said.
The Georgian authorities have released video footage which they say shows the Russian officers discussing military installations with Georgian citizens and exchanging money.
Georgia has also linked the four officers to an attack in the town of Gori which killed three police officers and injured dozens of people.
Russian Defence Minister Sergei Ivanov said the charges against the officers were false - and were intended to divert attention from Georgia's internal problems.
Georgian President Mikhail Saakashvili has denounced the Russian reaction to the arrests as "hysteria".
Relations between Moscow and Tbilisi have deteriorated in recent weeks, since Georgia and the Western military alliance Nato agreed to talks on closer relations, correspondents say.
The Russian foreign ministry said it decided to begin the evacuation of its personnel because of "a growing threat to their security".
It said emergency situations ministry planes would start flying Russian officials out of Georgia on Friday.
Moscow has also advised Russians not to visit Georgia, and has stopped issuing visas to Georgian nationals.
Georgia has accused Russia of actively trying to undermine its government by backing separatists in the breakaway regions of Abkhazia and South Ossetia.
Wednesday's arrests coincided with the first official visit by pro-Western President Saakashvili to a disputed area on the border between Georgia and Abkhazia.
Russia said his visit was dangerous and would raise tensions between Russia and Georgia.
Tbilisi also says Moscow is waging economic war against the country through embargoes on imports of Georgian products.
Russia has denied the allegations.
Relations between the two nations have become increasingly tense since Mr Saakashvili came to power in 2004, pledging to take the Caucasus nation out of Russia's orbit and join Nato and the European Union.
In 2005, Russia agreed to close its two remaining Soviet-era bases in Georgia by 2008.