(Adds Obama remarks, State Department comment)
* Four-way talks end with call for end to Ukraine violence
* Putin says Ukraine authorities heading for "abyss"
* Obama says talks offer hope that diplomacy can work
* Three separatists die in overnight shooting
By Arshad Mohammed and Alexei Anishchuk
GENEVA/MOSCOW, April 17 (Reuters) - The United States, Russia, Ukraine and the European Union called after crisis talks on Thursday for an immediate halt to violence in Ukraine, where Western powers believe Russia is fomenting a pro-Russian separatist movement.
President Barack Obama said the meeting in Geneva between Russia and western powers was promising but that the United States and its allies were prepared to impose more sanctions on Russia if the situation fails to improve.
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 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 wwere 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 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.
"There is the possibility, the prospect, that diplomacy may de-escalate the situation," Obama told reporters.
"The question now becomes, will in fact they use the influence they've exerted in a disruptive way to restore some order so that Ukrainians can carry out an election and move forward with the decentralization reforms that they've proposed," Obama said at the White House.
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.
"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.
The U.S. State Department said the talks had achieved more than some people had expected.
But a spokeswoman added: "But again, it's not a breakthrough until this is implemented on the ground, and we need to see the Russians follow up these words with actions."
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 State John Kerry said there would be additional sanctions on Russia if it did not act to calm tensions in Ukraine.
"If ... 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 Russians, 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 nations at least are reluctant to press ahead with more sanctions, fearing that could provoke Russia further or end up hurting their own economies.
Kerry also took the opportunity to condemn as "intolerable" suggestions in the eastern city of Donetsk that Jews had been ordered to register with authorities.
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, while the United States approved more non-lethal military support for Ukraine.
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.
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.
He also acknowledged for the first time that Russian troops had played a direct role in Crimea, assisting local militia.
The broadcast 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, David Brunnstrom and Mark Felsenthal in Washington; Writing by Peter Graff and Giles Elgood; editing by David Stamp)