Russia escalated its assault on Sunday despite strong diplomatic warnings from Mr. Bush and European leaders, underscoring the limits of Western influence over Russia at a time when the rest of Europe depends heavily on Russia for natural gas and the United States needs Moscow’s cooperation if it hopes to curtail what it believes is a nuclear weapons threat from Iran.

President Bush, in Beijing for the Olympics, strongly criticized the Russian attacks, especially those outside South Ossetia, and urged an immediate cease-fire.

In an interview on NBC on Monday morning, he said he had been “very firm” with both Russia’s prime minister, Vladimir V. Putin, and its president, Dmitri Medvedev.

Earlier, Vice President Dick Cheney expressed a strong warning for Russia. In a telephone conversation with the Georgian president, he said “that Russian aggression must not go unanswered, and that its continuation would have serious consequences for its relations with the United States, as well as the broader international community,” a spokeswoman, Lea Anne McBride, said in a statement released by the White House.

Russian officials say Georgia provoked the assault by attacking South Ossetia last week, causing heavy civilian casualties. But Western diplomats and military officials said they worried that Russia’s decision to extend the fighting and open a second front in Abkhazia indicated that it had sought to use a relatively low-level conflict in a conflict-prone part of the Caucasus region to extend its influence over a much broader area.

On Sunday, Russian artillery shells slammed the city of Gori, a major military installation and transportation hub in Georgia. In the separatist region of Abkhazia, Russian paratroopers and their Abkhaz allies battled Georgian special forces and tried to cross the boundary into undisputed Georgian territory, Georgian officials said.

Russia dropped a bomb on Tbilisi’s international airport shortly before Foreign Minister Bernard Kouchner of France, who was sent by the European Union to try to mediate, was due to land, Georgian officials said. It twice bombed an aviation factory on the outskirts of the capital. Russia’s Black Sea Fleet patrolled the coast of Abkhazia, and its Defense Ministry said Russian warships had sunk a Georgian gunboat that fired on them.