Rice, speaking at a joint news conference with Mikheil Saakashvili, the Georgian president, said: "It (Russia) needs to be a part of resolving the problem and solving the problem and not contributing to it. I have said it to the Russians publicly. I have said it privately.
"The violence needs to stop and whoever is perpetrating it, and I have mentioned this to the president, there should not be violence."
Rice also reaffirmed Washington's support for Georgia's bid to join Nato.
South Ossetia and Abkhazia broke away from Tbilisi's rule after wars in the 1990s, Russia says Tbilisi wants to use force to re-establish its control.
In the worst violence in months, a bomb in a cafe in Abkhazia killed four people on Sunday and separatists in South Ossetia said two people were killed last week in a heavy exchange of fire with Georgian forces.
Russia said there was strong evidence Georgia's government was behind the violence, though Tbilisi denied that.
Meanwhile, Russia confirmed on Thursday that its air force had flown over South Ossetia this week because it believed Tbilisi was preparing to attack the region.
Russia's foreign ministry said: "The need arose to take urgent and active measures to prevent bloodshed and keep the situation within peaceful bounds.
"In order to clarify the situation, aircraft of the Russian air force carried out a brief flight over the territory of South Ossetia.
|Russia said its jets had flown over South Ossetia earlier this week [GALLO/GETTY]
"As subsequent events showed, this step allowed [us] to cool hot heads in Tbilisi and prevent events developing along military lines, the likelihood of which was more than real."
Georgia has alleged in the past that Russia trespassed in its airspace but Moscow has alaways denied it.
The statement is Russia's first admission for at least a decade that its air force has flown over Georgian territory.
Georgia had earlier said Russian fighter jets entered its airspace late on Tuesday, less than 24 hours before Rice's visit.
Saakashvili said the jets had come close to the Georgian capital late on Tuesday.
"Maybe they wanted to salute Secretary Rice," he said. "I don't know ... This is a very worrisome development."
Saakashvili said Russia's behaviour was a reaction to Nato expansion and an increasing US presence in the region.
"Looks like some people did not notice that the Cold War is over. The main point is that Russia ... no longer acknowledges the jurisdiction of Georgia towards an essential part of its territory," he said.