A security officer and a pro-Russia activist were killed on Sunday during a raid launched by Ukrainian special forces in a city, where pro-Russia gunmen seized buildings of police and security services a day earlier, Xinhua reported.
Russian foreign ministry also called the Ukrainian acting President Alexandr Turchynov's order to use the armed forces for quelling protests outrageous.
"The Russian side is bringing up the crisis in southeastern Ukraine for urgent discussion at the UN Security Council and the Organisation for Security and Cooperation in Europe (OSCE)," the ministry said in an online statement.
Russia demanded that the Maidan organisers "who overthrew the legitimate president" immediately stop the war against their own people and fulfill all obligations under the Feb 21 agreement.
Meanwhile, Moscow urged the Maidan party's "Western sponsors" as well as the US to "subdue their unruly followers, make them distance themselves from the neo-Nazis and other extremists".
Russia called for an end to the use of the military against the Ukrainian people and the start of a national dialogue for the sake of early and radical constitutional reform in Ukraine.
A new wave of unrest erupted in Ukraine's east last weekend, when pro-Moscow activists seized several government buildings in the cities of Donetsk, Lugansk and Kharkov, demanding a referendum on autonomy and closer ties with Russia.
On March 18, Russian President Vladimir Putin and Crimean leaders signed a treaty to make the southern peninsula part of Russian territory after almost 97 percent of Crimean voters backed secession from Ukraine in a referendum.
Kiev has rejected the referendum and Crimea's integration with Russia, saying it was unconstitutional.