Georgia has accused Moscow of stalling and seeking to spread further into the country.
Speaking in Moscow, Lavrov said Russia did not intend to occupy Georgian territory and that Nato's encouragement to Georgia was "anti-Russian" in its intent.
He also said that Russian troops could be withdrawn from Georgia depending on how quickly Georgian troops returned to their permanent bases.
Speaking to Al Jazeera, Konstantin Kosachev, the head of Russia's foreign affairs committee in the Duma, Russia's parliament, said: "The most unpleasant truth in this story is that Nato has failed to prevent this conflict.
"It has failed to keep Saakashvili [Georgia's president] away from his plans to start this war, and has failed to save human lives. This is why Russia had to interfere and act the way Russia acted."
Jonah Hull, Al Jazeera's correspondent in Tbilisi, said: "One suspects that there is unfinished business here for the Russians.
"One of their main motivations has been to see the back of Saakashvili, Putin's long-time nemesis.
"It [the delay in retreating] is aimed at intimidating him and destabilising the country," Hull said.
Speaking on behalf of Nato on Tuesday, Jaap de Hoop Scheffer, the alliance's secretary general, said: "We cannot continue with business as usual".
He said Russian troops were not withdrawing from Georgia as promised in the six-point European Union-brokered ceasefire.
| De Hoop Scheffer urged Russia to honour its withdrawal pledge [AFP]
De Hoop Scheffer said: "They are not withdrawing as the Russian president promised."
"It's not business as usual. They're occupying a sovereign nation," he said.
The secretary general said Nato stood by its decision in Bucharest in April to allow Georgia to join Nato at some point in the future.
"Georgia can become a member of the alliance and in my opinion one day will become a member of it. The Bucharest decision stands."
Hull said the meeting had one simple intention "to call Russia to comply under the peace plan and to show Nato's full support for Georgia and its future membership plans".
"I don't think this is a very strong response and I don't think anyone in Moscow will think it's a strong response. Why would they bow at this point to a really not very aggressive response by Nato when they really don't need to?"
Condoleezza Rice, the United States secretary of state, who also attended the meeting in Brussels, warned Moscow that it was playing a dangerous game with the US and its Nato allies.
She said the West was determined to prevent Russia winning a strategic victory from its conflict with Georgia.
Rice said Russia's actions had been aimed at undermining Georgia's democracy, weakening its government and intimidating people there and in the region.
She said: "Russia is a state that is unfortunately using the one tool that it has always used ... when it wishes to deliver a message: that is its military power. We're determined to deny them their strategic objective."
A column of Russian tanks and armoured vehicles was said to have left the Georgian city of Gori heading towards Tskhinvali, the South Ossetian capital, on Tuesday in what Russian officials said part of the withdrawal demanded by the West.
However, Shota Utiashvilli, a spokesman for Georgia's interior ministry, said there had been no significant Russian troop movements.
He said Russian troops remained entrenched in previous positions, including in and around Gori.
"There is still no sign of a withdrawal, nothing at all," Utiashvilli said.
Russia's navy said on Tuesday it had cancelled a September visit by a US navy frigate to a port in Russia's Far Eastern region of Kamchatka.
The navy said in a statement: "At the present time it is not considered possible to accept the guided missile frigate USS Ford on a business visit."
The navy said the visit had been planned for 5-9 September.
The statement gave no further details about the reason for cancelling the visit and a Russian navy spokesman declined to comment further.
Last week, the US pulled out of a planned four-nation naval exercise with Russia in the Pacific because of Russia's actions in Georgia.
The conflict has been ongoing since Georgia attacked the breakaway province of South Ossetia on August 7.