Putin has rejected claims that Russian special forces are directing or encouraging the insurgents. Putin also has said he hopes not to send troops into eastern Ukraine, but he retains the right to intervene if necessary to protect ethnic Russians living here.
Russian state media have been feeding fears among the Russian-speaking population in eastern Ukraine that their lives are in danger because of the Right Sector.
"See what is happening?" asked Andrei Zarubin, 30, who came to the Bylbasivka checkpoint Sunday afternoon to replace those who had come under attack. "We hope that Russia will help, that Russia will protect us. Who else can we turn to for help?"
The Russian Foreign Ministry statement said the attack "proves the unwillingness of the Ukrainian authorities to restrain and disarm the nationalists and extremists."
After last week's talks in Geneva, top diplomats from Ukraine, Russia, the United States and the European Union called for an array of actions, including the disarming of militant groups and the freeing of public buildings taken over by insurgents.
Those terms quickly became a heated issue as pro-Russian armed groups that have seized police stations and other government buildings in eastern Ukraine said they would not vacate them unless the country's acting government resigned.
The insurgents say the Kiev authorities, who took power after pro-Russia Ukrainian President Viktor Yanukovych was ousted in February following months of protests, aim to suppress the country's Russian-speakers. Eastern Ukraine, which was Yanukovych's support base, has a substantial Russian-speaking population.
Russia also insists that the Kiev government should disarm members of the Right Sector, whose activists are occupying several buildings in the center of the capital, having turned them into makeshift offices.
Associated Press writer Lynn Berry in Moscow contributed to this report.