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Lavrov's comments came a day after US Vice President Joe Biden was in the Ukrainian capital with promises of support for the pro-Western government, and a warning to Russia not to interfere in Ukraine.
The crisis in Ukraine, now in its fourth month, has dragged Russia's relations with the West to their lowest since the Cold War. In the east, pro-Russian armed separatists have seized about a dozen public buildings and are defying Kiev's authority.
A further escalation could lead to damaging economic sanctions, and raises the risk of a disruption to the Russian gas supplies on which Europe depends.
NATO says Russia has built up a force of about 40,000 troop in its border with Ukraine. Moscow says some are stationed there permanently, while others have been deployed as a precaution to protect Russia from the instability in Ukraine.
In Moscow, Lavrov said Moscow would respond if its interests, or the interests of Russian citizens, were attacked.
"Russian citizens being attacked is an attack against the Russian Federation," he said according to excerpts of an interview with the Russia Today news channel.
"There is no reason not to believe that the Americans are running the show," RT quoted him as saying.
Washington said that suggestion was "ludicrous", while NATO's deputy secretary general, Alexander Vershbow, said Russia must de-escalate the situation and avoid "inflammatory rhetoric and misrepresentations of the situation inside Ukraine".
Russia justified its intervention in Crimea earlier this year by saying it had to defend Russians living there. In eastern Ukraine some people hold Russian passports.
Ukraine called on Moscow to pull troops back from the border, saying it feared pro-Russian separatists could use their proximity to provoke a Russian invasion.
Lavrov's ministry, in a separate statement, accused the United States and the interim government in Kiev of a "distorted interpretation" of an international accord, signed in Geneva last week, under which illegal armed groups in Ukraine are to disarm and give up buildings they have occupied.
Russia said that applies not only to the pro-Russian separatists in the east, but also to groups in Kiev whose protests helped bring Ukraine's new government to power.
Earlier, Ukraine's government relaunched a security operation to crack down on the pro-Russian armed groups after an Easter pause and said it had the backing of the United States.
But it was unclear what steps Kiev could take to restore its authority in the mainly Russian-speaking east, without wrecking the Geneva deal.
"The security forces are working on the liquidation of illegal armed groups," in the east of Ukraine, First Deputy Prime Minister Vitaly Yarema told reporters.
"The corresponding activities will be carried out in the near future, and you will see the results."
The Interior Ministry said it had flushed armed separatists out of a town which they had controlled in eastern Ukraine in an "anti-terrorism" operation.
It said the operation took place on the outskirts of the town of Sviatogorsk and that no one was injured. There had been no previous reports of gunmen in the town, which lies just outside the stronghold of pro-Russian militants in Slaviansk.
Ukraine's SBU state security service warned that it would attack militants who held out. It said the Geneva accord required all illegal militias to lay down their arms:
"If not, the law enforcement agencies will use all their forces, means and capabilities to put an end to criminal activities and restore law and order and communal security."
Kiev's decision to resume its security operation in the east was prompted in part by the discovery of two bodies in a river in eastern Ukraine. One body was that of Volodymyr Rybak, a member of the same party as Ukraine's acting president.
The Ukrainian government, which took power after Moscow-backed President Viktor Yanukovich fled the capital in a row over whether to strengthen ties with Europe, appeared emboldened by Biden's visit on Tuesday.
He brought a package of aid and urged Russia to curb the separatist militias in the east.
"We have obtained the support of the United States, that they will not leave us alone with an aggressor. We hope that in the event of Russian aggression, this help will be more substantive," Yarema said.
The United States and NATO have made clear they will not intervene militarily in Ukraine. But the Pentagon said it was sending about 600 soldiers to Poland and the three Baltic states for infantry exercises, to reassure NATO allies.
Russian gas giant Gazprom has said it will turn off supplies to Ukraine next month unless Kiev pays its outstanding debts. That would have a knock-on effect on deliveries to Europe, because much of the gas shipped westwards has to pass through Ukrainian territory.
The European Commission said it would meet Slovakian and Ukrainian ministers on Thursday to discuss the possibility of pumping gas back to Kiev. Another meeting between the Commission, Ukraine and Russia is due on Monday on Moscow.
The crisis in Ukraine began when Yanukovich, under pressure from Moscow, pulled out of a planned cooperation agreement with the EU. Pro-Western protesters took to the streets and Yanukovich fled after bloody clashes.
As a caretaker leadership of pro-Western protest leaders took over the government in Kiev, the Kremlin sent its forces into Ukraine's Crimea Peninsula, and shortly after annexed the region. Moscow said it acted to protect local people who were being persecuted by Kiev's new rulers, while the West called it an illegal land grab.
Mediators from the Organisation of Security and Cooperation in Europe, tasked with helping the sides implement the accord, were in eastern Ukraine trying to encourage illegal groups to disarm. There was no sign yet they were backing down.
In areas under the separatists' control, there was growing evidence of arbitrary rule by self-appointed local officials, backed up by heavily armed militias, and of violence being meted out against opponents.
A video released on a local news site, gorlovka.ua, purported to show Rybak, the councillor whose body was found in a river, being confronted by an angry crowd outside the town hall in Horlivka, where he was a councillor.
Rybak can be seen being manhandled by several men, among them a masked man in camouflage, while other people hurl abuse.
After several minutes, Rybak appears able to walk away. The Interior Ministry said he was seen being bundled into a car by masked men in camouflage later that day. His body, and that of a second man, was found on Saturday in a river near Slaviansk.
Ukraine's SBU security service issued a statement accusing a rogue SBU officer and an officer in Russia's GRU military intelligence of involvement in Rybak's murder.
In nearby Slaviansk, pro-Russian militia were holding three journalists, including a US citizen, Simon Ostrovsky, from the online news site Vice News.
The United States said the detentions amounted to kidnappings which violated the Geneva agreement.