(Adds Rice comments, U.S. envoy's name, details)

WASHINGTON, Aug 8 (Reuters) - The United States told Russia on Friday to withdraw its forces from U.S. ally Georgia, stop its air attacks and respect Georgia's territorial integrity, following fighting in the breakaway region of South Ossetia.

"We call on Russia to cease attacks on Georgia by aircraft and missiles, respect Georgia's territorial integrity, and withdraw its ground combat forces from Georgian soil," U.S. Secretary of State Condoleezza Rice said in a statement.

Rice issued her statement as Georgia, a former Soviet state that now wants to join NATO, battled to get control of the rebel enclave backed by Russian forces.

Both Rice and the White House urged an immediate cease-fire in South Ossetia, and U.S. officials said they would send an envoy to the region to help mediate.

But as fighting continued to rage in and around the South Ossetian capital of Tskhinvali and Georgia's President Mikheil Saakashvili said Russia and Georgia were at war, Rice went further and addressed the Russian role directly.

She said the United States was working actively with its European partners to launch international mediation, and "we urgently seek Russia's support of these efforts."

Diplomatic sources said the U.S. envoy would be Deputy Assistant Secretary of State for European and Eurasian Affairs, Matthew Bryza, who is expected to join a mission that includes the European Union and the Organisation for Security and Cooperation in Europe.

Georgia, which hosts oil pipelines from the Caspian Sea to world markets, is at the center of a battle for influence between the United States and Russia in the Caucasus.

President George W. Bush has called Georgia a "beacon of democracy" in a volatile region, and sent Rice to Tbilisi last month partly to encourage resolution of its dispute with Russia over the separatist regions of South Ossetia and Abkhazia.

Bush was in Beijing, where he discussed the crisis with Russian Prime Minister Vladimir Putin on the sidelines of the Olympics. Moscow says its troops were responding to a Georgian assault to retake South Ossetia after the collapse of a brief cease-fire on Thursday.

NATO MEMBERSHIP

The Pentagon said it was monitoring events but has received no request for assistance from Georgian officials.

Washington has pushed hard for NATO membership for Georgia, despite European misgivings over the state's stability. NATO membership would bring Georgia under the organization's mutual defense umbrella in which an attack on one member is considered an attack on all.

Russia opposes Georgia's NATO bid and accuses Georgia of fueling tension. Moscow says its role has been to defend people in South Ossetia and Abkhazia from Georgian aggression.

Robert Hunter of the RAND Corporation, a former U.S. ambassador to NATO, said "no one wants to fight for Georgia."

He said the United States had pushed too hard to try to get NATO membership for Georgia, and now Russia was trying to demonstrate, in part, that U.S.-led NATO efforts had their limits and Moscow's interests must be taken into account.

Republican presidential candidate John McCain said Russia should pull its forces out. "The consequences for Euro-Atlantic stability and security are grave," he said.

Democratic candidate Barack Obama said he "wholeheartedly condemned" the violation of Georgia's sovereignty. "What is clear is that Russia has invaded Georgia's sovereignty, has encroached on Georgia's sovereignty, and it is very important for us to resolve this issue as quickly as possible."

The U.S. embassy in the Georgian capital of Tbilisi issued a warning to Americans to defer "all travel to the conflict zones" of South Ossetia and Abkhazia and avoid nonessential travel inside Georgia.

Janusz Bugajski of the Center for Strategic and International Studies said the United States could not abandon a pro-American country that gave "enormous" amounts of intelligence about terrorist operations across the Caucasus and sent more troops to Iraq, per capita, than other nations.

Georgia announced it was withdrawing 1,000 soldiers from Iraq to help fight off Russian forces in South Ossetia.

If the crisis escalates, some U.S. military aid would be essential, such as logistics or intelligence, Bugajski said. "We cannot see Georgia disappear under the Russian boot." (Additional reporting by David Morgan, Caren Bohan, Jeremy Pelofsky and Matthew Bigg; editing by Mohammad Zargham)