(Adds outcome of talks )
* Four-way talks end with call for end to Ukraine violence
* Putin says Ukraine authorities heading for "abyss"
* Three separatists die in overnight shooting
GENEVA/MOSCOW, April 17 (Reuters) - The United States, Russia, Ukraine and European Union together called on Thursday for an immediate halt to violence in Ukraine, where Western powers believe Russia is fomenting a pro-Russian separatist movement.
Washington immediately warned Moscow that it would face further sanctions if it did not carry out the agreement, reached in four-party crisis talks in Geneva.
Russian President Vladimir Putin, speaking in Moscow, accused Ukraine's leaders of committing a "grave crime" by using the army to try to quell unrest in the east of the country, and did not rule out sending in Russian troops.
Putin said he hoped he would not need to take such a step, and that diplomacy could succeed in resolving the standoff, the worst crisis in East-West relations since the Cold War.
The comments came hours after separatists attacked a Ukrainian national guard base overnight and Kiev said three of them were killed in the worst bloodshed yet in a 10-day pro-Russian uprising.
Ukrainian, Russian and Western diplomats held emergency talks seeking to resolve a confrontation that has seen pro-Russian fighters seize official buildings across eastern Ukraine while Moscow masses tens of thousands of troops on the frontier.
"All sides must refrain from any violence, intimidation or provocative actions," a joint statement issued after the Geneva talks said.
"All illegal armed groups must be disarmed; all illegally seized buildings must be returned to legitimate owners; all illegally occupied streets, squares and other public places in Ukrainian cities and towns must be vacated," it added.
"It will be a test for Russia, if Russia wants really to show willing to have stability in these regions," said Ukrainian Foreign Minister Andriy Deshchytsia.
It was unclear if Russia would meet Western demands for it to stop stirring unrest in the east and withdraw its troops from the Ukrainian border. Moscow denies it is active in Ukraine.
There was scepticism over whether the agreement could work.
"Diplomacy cannot succeed if there is no room for compromise," said Ulrich Speck, a visiting scholar at Carnegie Europe. "The Kremlin is dedicated to get Ukraine under its control, one way or another. It feels that it has well advanced on that goal, and is not ready to back down. The West simply cannot agree to those conditions."
U.S. Secretary of States John Kerry said there would be additional sanctions on Russia if it did not act to calm tensions in Ukraine.
"If there is not progress over the course of these next days, and we don't see a movement in the right direction, then there will be additional sanctions, additional costs as a consequence," Kerry told reporters.
The United States and European Union have so far imposed visa bans and asset freezes on a small number of Russian individuals, a response that Moscow has openly mocked. However, the Western states say they are now contemplating measures that could hurt Russia's economy more broadly.
But some EU states at least are reluctant to press ahead with more sanctions, fearing that could provoke Russia further or end up hurting their own economies.
Kerry and European Union foreign policy chief Catherine Ashton said the United States and the EU still had a significant difference with Russia over the status of Ukraine's Crimea region, annexed by Moscow last month.
Moscow's takeover of the Black Sea peninsula followed the overthrow of Ukraine's pro-Moscow president, Viktor Yanukovich, after months of street protests prompted by his rejection of a trade deal with the EU.
Seeking to reassure its eastern allies, NATO announced it was sending warships to the Baltic. The United States approved more non-lethal military support for Ukraine, including medical and welfare supplies.
Speaking on Russian television Putin accused the authorities in Kiev of plunging the country into an "abyss".
The Kremlin leader overturned decades of post-Cold War diplomacy last month by declaring Russia had a right to intervene in neighbouring countries and by annexing Crimea.
Kiev fears he will use any violence as a pretext to launch an invasion of eastern Ukraine by Russian forces.
"Instead of realising that there is something wrong with the Ukrainian government and attempting dialogue, they made more threats of force ... This is another very grave crime by Kiev's current leaders," Putin said in his annual televised question-and-answer session with the Russian public.
"I hope that they are able to realise what a pit, what an abyss the current authorities are in and dragging the country into," said Putin.
Ukraine, which also sees Moscow's hand behind separatist uprisings in the east, said it would impose stricter border controls on Russian men seeking to enter the country. Russia said it might retaliate.
At the Ukrainian national guard headquarters in Mariupol, on the Sea of Azov, there was evidence the building had come under attack.
A grey police jeep was inside the compound on Thursday morning with broken windows, flat tyres and bent doors. The gates of the compound had been flattened. There were shell casings outside the gates and several unused petrol bombs.
Interior Minister Arsen Avakov said an armed group of about 300 separatists attacked the base with guns and petrol bombs. Three separatists were killed and 13 wounded, he said.
Putin's televised chat, in a talk show format with satellite link-ups and applauding audiences across Russia, lasted for several hours. His words were clearly directed both at a domestic audience and at a world still grappling with the implications of his new doctrine, which the West says dispenses with customary limits on the use of armed force.
A self-confident Putin pointed to authorisation he secured in March from the mostly appointed upper house of parliament to use force in Ukraine, though he said he preferred negotiations.
"The Federation Council granted the president the right to use military force in Ukraine. I really hope that I do not have to exercise this right and that we are able to solve all today's pressing issues via political and diplomatic means," Putin said.
Putin cast the Crimea annexation in explicitly geo-political terms, saying it was a response to the prospect that NATO could one day incorporate Ukraine and cut off Russia's access to its Black Sea Fleet. Previously the operation had been justified solely as reflecting the will of Crimea's residents.
He also acknowledged for the first time that Russian troops had played a direct role in Crimea, assisting local militia.
Putin's chat even featured a cameo appearance from Edward Snowden, the former U.S. security contractor given asylum in Russia after leaking information about surveillance by U.S. and British spy agencies. Snowden, patched in by video link, asked a question about Russian surveillance. Putin denied that Moscow carried out mass collection of citizens' data.
Pro-Russian militants control buildings in about 10 towns in eastern Ukraine after launching their uprising on April 6.
Separatists occupying a local government building in the city of Donetsk said they would not leave until supporters of Ukraine's new government quit their camp around Kiev's main square, known as the Maidan.
Asked how his group will react to the accord in Geneva under which illegal occupations of buildings and squares must end, Alexander Zakharchenko, a protest leader inside the Donetsk regional government building, told Reuters by telephone:
"If it means all squares and public buildings, then I guess it should start with the Maidan in Kiev. We'll see what they do there before we make our decision here."
(Additional reporting by Richard Balmforth in Kiev, Stephanie Nebehay, Arshad Mohammed and Catherine Koppel in Geneva, Christian Lowe and Alexei Anishchuk in Moscow and Jeff Mason in Washington; Writing by Peter Graff and Giles Elgood; editing by David Stamp)