Georgian television showed a bomb-damaged car and police officers standing nearby.
Alexander Diordiev, assistant commander of the Russian peacekeeping force in Abkhazia, was quoted as saying by Interfax news agency that the four blasts happened "almost simultaneously".
Lieutenant-Colonel Alexander Novitsky, a spokesman for the peacekeepers, was quoted by Russian news agencies as blaming the blast on Georgian agents.
The RIA news agency reported Novitsky as saying a Georgian special forces uniform had been found wrapped around the shell that caused one of the blasts.
"In the place of the explosion, there remained the uniform," he was quoted as saying.
Utiashvili denied the allegations and said that separatist forces had shelled Georgian villages in the border area.
Meanwhile, a town in nearby South Ossetia also reportedly came under fire from the Georgian side of the border on Saturday, according to Russia's Vesti-24 news channel.
A spokesman for the South Ossetian side in the joint commission on the crisis resolution said that a police post came under fire from guns and grenade launchers but there were no reports of injuries.
Tensions in the two breakaway regions have risen since Russia earlier this year moved to bolster ties with them.
Abkhazia and South Ossetia have strong diplomatic and economic support from Russia, while Tbilisi has accused Moscow of attempting to annex the provinces.
Abkhazia's separatist government and Russian peacekeepers have both accused Tbilisi of planning military operations to retake the region which gained de facto independence after a conflict in the mid-1990s.
On Saturday, Dmitry Medvedev, Russia's president, urged his Georgian counterpart Mikheil Saakashvili to refrain from "stoking tensions" in the two regions.
Georgia has accused the Russian peacekeeping forces of supporting the separatist governments and secretly bringing artillery and other heavy weapons into the breakaway regions.