* Pro-Russia separatists say they came under gun attack
* Russia, Kiev question each other's compliance with peace accord
* Washington says more sanctions on Russia if no progress
* Reportedly vacated town hall is still occupied
* U.S. Vice President Biden visits Ukraine this week (Adds Biden to visit Ukraine, paragraphs 5-6)
By Aleksandar Vasovic and Alissa de Carbonnel
SLAVIANSK/YENAKIEVO, Ukraine, April 20 (Reuters) - At least three people were killed in a gunfight in the early hours of Sunday near a Ukrainian city controlled by pro-Russian separatists, shaking an already fragile international accord that was designed to avert a wider conflict.
The incident triggered a war of words between Moscow and Ukraine's Western-backed government, with each questioning the other's compliance with the agreement, brokered last week in Geneva, to end a crisis that has made Russia's ties with the West more fraught than at any time since the Cold War.
The separatists said armed men from Ukraine's Right Sector nationalist group had attacked them. The Right Sector denied any role, saying Russian special forces were behind the clash.
Failure of the Geneva agreement could bring more bloodshed in eastern Ukraine, but may also prompt the United States to impose tougher sanctions on the Kremlin - with far-reaching consequences for many economies and importers of Russian energy.
U.S. Vice President Joe Biden will meet with Ukraine's acting president, Oleksander Turchinov, and Prime Minister Arseny Yatseniuk in Kiev on Tuesday during a two-day stay in Ukraine.
Biden will consult on developments in eastern Ukraine and discuss international efforts to strengthen Ukraine's economy and assist Kiev "in moving forward on constitutional reform, decentralization, anti-corruption efforts, and free and fair presidential elections on May 25th," the White House said on Sunday.
Under the deal signed in Geneva last week, the European Union, Russia, Ukraine and the United States agreed that illegal armed groups would go home in a process to be overseen by Europe's OSCE security watchdog.
So far, the pro-Russian militants have shown little sign of budging from public buildings in the east, although there was some hope of progress after Kiev said it would not move against the separatists over Easter, and international mediators headed to eastern Ukraine to try to persuade them to disarm.
But the shootings near Slaviansk - already a flashpoint for tensions between Ukraine's rival camps - are likely to make that task even harder, hardening the view of the many Russian speakers in eastern Ukraine that they cannot trust Kiev.
"The Easter truce has been violated," the Russian Foreign Ministry said in a statement. "This provocation ... testifies to the lack of will on the part of the Kiev authorities to rein in and disarm nationalists and extremists."
The town's self-appointed pro-Russia mayor placed a curfew on the town and appealed directly to Russia's Vladimir Putin to consider sending in peacekeeping troops - an outcome Ukraine tried to avoid by holding back its poorly resourced forces.
Ukraine's SBU security service accused Moscow agents of faking a "cynical provocation" at Slaviansk and the Foreign Ministry hit back, reproaching Russia for rushing to judgment and failing to meet its part of the deal struck in Geneva.
"The Russian side must be reminded about their obligations under the Geneva agreement to bring all necessary influence to bear on separatists to clear illegally held buildings, unblock roads, lay down arms and prevent any bloodshed," it said.
Right Sector spokesman Artem Skoropadsky said it was a "blasphemous provocation from Russia: blasphemous because it took place on a holy night for Christians, on Easter night. This was clearly carried out by Russian special forces."
Separatist militiamen near the eastern Ukrainian city of Slaviansk told Reuters four vehicles had approached their checkpoint at around 2:00 a.m. (2300 GMT) and opened fire.
"We had three dead, four wounded," one of the separatist fighters, called Vladimir, told Reuters at the checkpoint, where there were two burned-out jeeps.
He said the separatists returned fire and killed two of the attackers, who he said were members of the nationalist movement that has its power base in the Ukrainian-speaking west of the country and is reviled by many in the Russian-speaking east.
Police in Kiev said three men among the separatists were killed and three wounded.
A Reuters cameraman at the scene said he saw the bodies of two people, one with what appeared to be gunshot wounds to the head and face, lying in the back of a truck.
One of the dead was dressed in camouflage fatigues, the other, identified by several bystanders as a local man, was in civilian clothes.
The deaths were the first in armed clashes in eastern Ukraine since the Geneva accord was signed on Thursday.
NO WILL TO MOVE
The crisis in Ukraine began late last year when President Viktor Yanukovich turned his back on closer ties with Europe, prompting protests in the capital. They led to him fleeing Kiev, and a pro-Western interim administration taking over.
Soon after, Moscow used its military to back separatists in Ukraine's Crimea, before Putin signed a document annexing the peninsula. The United States and European Union responded by slapping sanctions on Russian officials.
A local media report that in Yenakievo, Yanukovich's home town outside Donetsk, separatists had left the town hall they had occupied for a week prompted the U.S. ambassador to Ukraine to speak of "progress". But a Reuters correspondent later found the building flying the flag of the separatist Donetsk Republic.
One activist there said they had not ended the occupation. Local police were in position outside the town hall.
Russia, which has deployed extra troops on the border with Ukraine, says it is not interfering but has an obligation to protect the Russian-speaking community in the east from the rulers in Kiev, who it says are illegitimate, have ties to the far right, and discriminate against ethnic Russians.
Senior officials from the OSCE, in Donetsk to mediate an end to the crisis, cautioned against expecting rapid results.
"This will take time," German diplomat Klaus Zillikens told Russia's Ekho Moskvy radio station. "Of course, the ideal outcome for everyone would be if all sides agreed and straight away said: 'OK, let's implement the agreement'. But it won't be like that. And we can see that already."
The Ukrainian Foreign Ministry had promised that, as a gesture of goodwill for the Easter holiday, it would suspend the "active phase" of an operation it had launched to reassert its authority in the east of the country - although in truth there had been little sign of its limited forces making much impression.
In another sign of reconciliation, the Interior Ministry issued an Easter message that asked members of the ministry's disbanded Berkut unit to help defend Ukraine's unity.
In the days when Yanukovich was clinging to power in Kiev, Berkut members are alleged to have been responsible for shooting dead dozens of protesters. Their unit was disbanded, and some of them have joined the pro-Russian separatists.
The separatists say they will not leave the buildings they have occupied until the pro-Kiev protest groups occupying Independence Square - scene of the months-long protests against Yanukovich, also go home. Moscow has supported that point of view.
The eastern activists want guarantees they will be given a large degree of autonomy from Kiev and that protections of Russian-language rights will be enshrined in a new constitution. The acting president, Turchinov, repeated his assurances on Sunday that power could be devolved within months.
Interviewed in Russian on a channel focused on the Russian-speaking east, he said he was ready to listen to local leaders and immediately appoint regional governors of their choosing.
Accusing Putin of destabilising Ukraine in order to control it, he said the Kremlin leader was afraid of the example Ukraine's uprising set for Russia and other ex-Soviet states.
At Easter church services in Kiev and in Moscow, senior clergy issued sharply contrasting appeals for peace.
"In these Easter days, our prayers to God are for the people of Ukraine, for a reconciliation of enmity, for an end of violence, for people's love for each other, so that they should not be divided," Patriarch Kirill, head of the Russian Orthodox Church, said in a recorded video message.
In his Easter message, Patriarch Filaret, head of Ukraine's Orthodox Church, condemned what he described as Russia's aggression against his country: "God cannot be on the side of evil, so the enemy of the Ukrainian people is condemned to defeat," he said. "Lord, help us resurrect Ukraine." (Additional reporting by Richard Balmforth, Natalia Zinets and Alastair Macdonald in Kiev, Dmitry Madorsky in Slaviansk, Alissa de Carbonnel in Donetsk, Conor Humphries and Christian Lowe in Moscow, and Jeff Mason in Washington; Writing by Christian Lowe and Richard Balmforth; Editing by Philippa Fletcher, Anna Willard and Peter Cooney)