Rice made the remark at a joint press conference in Tbilisi with Georgian President Mikheil Saakashvili. Their talks came amid rising tensions between Tbilisi and Moscow over Georgia's breakaway regions of Abkhazia and South Ossetia.
America's top diplomat traveled to Tbilisi just as tensions between Russia and Georgia took another turn for the worse.
U.S. Secretary of State Condoleezza Rice arrived in the Georgian capital on July 9 as Russia and Georgia were accusing each other of violating cease-fire agreements by flying jets over South Ossetia.
And Rice's visit came just days after the simmering conflict in another Georgian breakaway region turned bloody, with an explosion in Abkhazia that left four people dead.
Dialogue Only Route
During a news conference with Georgian President Mikheil Saakashvili on July 10, Rice said that dialogue was the only way to resolving differences among those involved.
"The violence needs to stop, and whoever is perpetrating it -- I've mentioned this to the president (Saakashvili) -- there should not be violence," Rice said. "Innocent lives have been taken, and it is very important that all parties reject violence as an option. There must be a peaceful solution to the situation in Abkhazia."
Rice added that finding such a solution would require discussion at the highest levels of government -- and not only between Russia and Georgia, but also between Georgia and its breakaway republics.
"I am going to call for discussions to be held at a higher level, at least at the level of political directors and perhaps beyond that, because this is a serious situation," she said. "There are ideas about de-escalation. There are ideas about ending the isolation of Abkhazia. Georgians and Abkhaz need to talk directly. So, there are many things that could be done, and we are going to pursue them, but I do think that we need to do it more intensively and at a higher level."
The U.S. secretary of state also reminded Russia of its responsibilities as a mediator in the disputes between Georgia and its breakaway regions, which Moscow supports.
"Russia is a member of the Friends [of the UN secretary-general for Georgia]. It really needs to behave in that way, which means that it needs to be a part of resolving the problem and solving the problem, and not contributing to it."
Abkhazia, in particular, has in recent months become a symbol of discord between Georgia and Russia. Moscow and Tbilisi have verbally sparred over the rise in the number of Russian troops in the breakaway region, and the overflights and downings of unmanned Georgian reconnaissance aircraft.
'This Is Their Reaction'
During the July 10 press conference with Rice, Saakashvili made clear that he believes Russia is encouraging instability in the region.
"The way the Russians have been explaining this publicly, as well as privately, to us is this is their reaction to NATO expansions plans, this is their reaction to the independence of Kosovo, and this is their reaction to the increasing U.S. presence here in the region," Saakashvili said. "It looks like some people have not noticed that the Cold War is over."
But the Georgian president, who has twice met with Russia's new president, Dmitry Medvedev, in recent weeks, also stressed the importance of traditional ties between Moscow and Tbilisi.
It looks like some people have not noticed that the Cold War is over.
"We have very strong ties with Russia," he said. "We have very strong emotional, cultural, historical ties, and we do certainly feel obliged for ourselves and for future generations to reestablish and forge very strong links with the Russians. At the same time, we are a small country, but we have national security interests and national interests."
Finding such common ground could be essential just to get all the various parties to sit down at the negotiating table.
But as Georgian National Security Council Secretary Alexandre Lomaia told RFE/RL's Georgian Service in an interview on July 10, the result should be win-win for all involved.
"Today, it's hard to talk about 'optimism,' but our attitude is quite positive, because the [peace] plan under discussion gives a chance to all involved parties -- I repeat, all involved parties, and these are the separatist authorities, the Georgian central government, Russia, EU, and the U.S. -- huge pluses, and there will practically be no losers here," he said.
Rice's visit to Tbilisi came on the heels of a trip to Prague in which an agreement was signed on the future construction of part of a U.S. antimissile shield on Czech soil.
Russian President Medvedev on July 9 said the Kremlin was "extremely distressed" by the development.
"Of course, we are not going to get hysterical about it," Medvedev added at the end of a G8 summit in Japan, "but we will think of retaliatory steps."
On the day of the signing, July 8, the Russian Foreign Ministry issued a statement saying that the retaliation would not follow a course of diplomacy, rather that of "military-technical methods."
Of course, we are not going to get hysterical about it, but we will think of retaliatory steps.
While in Bulgaria on July 9 -- the second day of her three-day trip to Central and Eastern Europe -- Rice shrugged off potential negative Russian reaction to her upcoming talks in Tbilisi.
"I don't expect much comment about the United States going to visit a friend," Rice said.
Indeed, her time spent in the Georgian capital featured a private dinner with the Georgian president and scenes of the two engaging in an easygoing back-and-forth during their joint press conference.
But Georgian State Reintegration Minister Temur Iakobashvili said there was much more behind the secretary of state's trip.
"I think this was not just a symbolic visit," he said. "This visit was aimed at the concrete result in three major directions: conflict resolution and Georgia's NATO integration, as well as development of democratic processes in Georgia. These are the three guiding themes which have accompanied the visit and represent priorities both for our country and the U.S. policy in this region."
RFE/RL's Georgian Service contributed to this report