* Operation underway in eastern Donetsk region - president
* Rebels continue occupation in quiet Slaviansk
* Ukraine on brink of civil war - Medvedev
* Kiev makes step in right direction - Lavrov
* U.S. and Russia tell each other to calm crisis
By Richard Balmforth and Thomas Grove
KIEV/SLAVIANSK, Ukraine, April 15 (Reuters) - Russia declared Ukraine on the brink of civil war on Tuesday as Kiev said an "anti-terrorist operation" against pro-Moscow separatists was underway, though the crackdown appeared to get off to a slow start, if at all.
Twenty-four hours after an Ukrainian ultimatum expired for the separatists to lay down their arms, witnesses reported no signs yet that Kiev forces were ready to storm state buildings in the Russian-speaking east that the rebels have occupied.
Interim president Oleksander Turchinov insisted the operation had started in the eastern Donetsk region, although it would happen in stages and "in a considered way".
Amidst the deepest East-West crisis since the Cold War, the leaders of Russia and the United States called on each other to do all in their power to avoid further bloodshed.
Russian Prime Minister Dmitry Medvedev gave a gloomy assessment after at least two people died on Sunday when Kiev unsuccessfully tried to regain control in Slaviansk, one of about 10 towns and cities where the separatists have seized buildings.
"Blood has once again been spilt in Ukraine. The country is on the brink of civil war," he said in a statement posted on his Facebook page.
Ukraine has accused Russia of stirring up the separatists following its annexation of Crimea, while Moscow says Kiev has provoked the crisis by ignoring the interests of its citizens who use Russian as their first language.
Turchinov said the offensive, which he first announced on Sunday, was finally underway. "The anti-terrorist operation began during the night in the north of Donetsk region. But it will take place in stages, responsibly, in a considered way. I once again stress: the aim of these operations is to defend the citizens of Ukraine," he told parliament.
On Tuesday morning a Reuters correspondent in Slaviansk had heard no shots or explosions in the town, which lies about 150 km (90 miles) from the Russian border.
Outside the occupied local police headquarters about a dozen civilians manned barricades that have been built up overnight with more tyres and wooden crates. A dozen or so armed Cossacks - paramilitary fighters descended from Tsarist-era patrolmen - stood guard at the mayor's offices. Shops were functioning normally, bread supplies normal.
"The night passed quickly, thank God. There have been lots of rumours of violence, but it's been very quiet here. We are in control," said one civilian on the barricades outside the police station, who gave his name only as Rustam.
A STEP IN THE RIGHT DIRECTION
Western officials have drawn parallels between events in the east of Ukraine and what happened in Crimea, which Russian troops seized in February before the formal annexation.
However, Russian Foreign Minister Sergei Lavrov said he welcomed signs that the government in Kiev was ready for dialogue with the separatists, Interfax news agency reported.
Speaking during a visit to Beijing, Lavrov said Kiev's apparent willingness to "resolve through negotiations all the problems relating to the legal demands of the inhabitants of the south-east regions of Ukraine, is certainly a step in the right direction, albeit very belated".
U.S. President Barack Obama criticised Russia in a telephone call with President Vladimir Putin on Monday, saying its actions were not conducive to a diplomatic solution.
"The president emphasised that all irregular forces in the country need to lay down their arms, and he urged President Putin to use his influence with these armed, pro-Russian groups to convince them to depart the buildings they have seized," the White House said in a statement.
Moscow put the onus for peace on Washington. "President Putin called on Barack Obama to do his utmost to use the opportunities that the United States has to prevent the use of force, and bloodshed," the Kremlin said in a statement.
NATO states have sent troops, aircraft and ships to eastern Europe to reassure nervous post-communist alliance members, including the Baltic states - which were once Soviet republics - as well as Romania and Bulgaria.
A Russian fighter aircraft made repeated low-altitude, close-range passes near a U.S. ship in the Black Sea over the weekend, the Pentagon said, condemning the action at a time of heightened U.S.-Russian tensions.
"This provocative and unprofessional Russian action is inconsistent with their national protocols and previous agreements on the professional interaction between our militaries," Colonel Steve Warren, a Pentagon spokesman, said on Monday.
The White House has warned Russia it would face further costs over its actions in Ukraine, but it made clear on Monday that the United States was not considering lethal aid for Ukraine.
European foreign ministers agreed to step up sanctions, and U.S. officials have said they were in consultations with European partners on how to punish Moscow for what Kiev and its Western allies call a Russian plot to dismember Ukraine.
Moscow has largely brushed off sanctions so far, which the United States and Europe have explicitly designed to target only a limited number of officials and avert wider economic harm.
In a bid to undercut the rebels' demands, Turchinov held out the prospect of a countrywide referendum on the future shape of the Ukrainian state. Pro-Russian secessionists want separate referendums in their regions, which Kiev says is illegal. (Additional reporting by Natalia Zinets in Kiev, Alessandra Prentice in Moscow, Missy Ryan, Jeff Mason and Arshad Mohammed in Washington; Writing by David Stamp; Editing by Will Waterman)