Russia withdrew its troops from the "buffer zones" last week
Georgia and Russia have blamed each other after an effort to bring them together in Geneva for talks about their war over South Ossetia failed.
The UN, EU and OSCE were hosting what would have been the first low-level talks between the two states since the conflict in early August.
The rival delegations did not formally meet during the day.
An EU official said the talks had been suspended until 18 November because of "procedural difficulties".
The talks had been aimed at encouraging security in the Caucasus, following the truce between Moscow and Tbilisi.
In August, Russia ousted Georgian troops trying to regain South Ossetia and it later recognised both that region and Abkhazia, another breakaway Georgian territory, as independent states.
In a separate development, the International Court of Justice has ordered Georgia and Russia to protect civilians from ethnic discrimination in both regions.
"The Russians and the Georgians were not in a formal meeting at the same time, they weren't in the same room at the same time," US Assistant Secretary of State for European and Eurasian Affairs Daniel Fried told reporters after the talks move failed.
EU representative Pierre Morel said there had been a "problem of status, format".
He added: "I won't go into details. All these, I think we can group them together under the question of procedure."
Russia insists on having Abkhaz and South Ossetian delegates attend any talks, while Georgia refuses to recognise their secession.
In Brussels, Georgian President Mikhail Saakashvili accused Russia of "walking out" of the talks.
"[This] basically means that Russia has no interest whatsoever at this stage in any diplomatic process," he said.
But the head of Russia's delegation, Deputy Foreign Minister Grigory Karasin, said Mr Saakashvili's description of events was "just a lie".
"The Georgian version just doesn't hold water," he told a briefing. "We deplore the absence of the Georgian delegation but we did not see it as tragic."
It was a surreal, almost farcical day in Geneva, the BBC's Imogen Foulkes reports from the city.
She says that after a big build-up in which UN Secretary General Ban Ki-Moon, French Foreign Minister Bernard Kouchner, and the chair of the Organisation for Security and Co-operation in Europe (OSCE), Alexander Stubb, all arrived in town to say how wonderful it was that the talks were taking place, the discussions ended after just a couple of hours.
Russian and Georgian claims that the other side did not turn up were a difficult conundrum for OSCE mediator Heikki Talvitie to explain, our correspondent notes.
"Let's put it in a way that we have two meetings," said Mr Talvitie.
"One formal, one informal, and all participants were in one or the other meeting. We are not negotiating face to face we are discussing. We are not going to make decisions."
The UN and the EU are clutching hopefully to the fact that a further meeting is set for November but after this outcome it is hard to believe it is worthwhile, our correspondent says.
One of the main items on Wednesday's original agenda - the future of those displaced by the conflict - was never even discussed, she adds.