"The Georgian secret services participated in organising these explosions in Gagra and Sukhumi," Bagapsh said.
"The security services of Abkhazia will do everything possible to find those responsible."
Tbilisi has denied any involvement in the attacks, which Abkhaz officials say were carried out to damage tourism in the region.
"We had absolutely nothing to do with the recent explosions in Abkhazia," Shota Utiashvili, a Georgian interior ministry spokesman, said.
The differences between Georgia and Abkhazia come amid tensions Tbilisi and its neighbour Russia.
Tbilisi has accused Russia of supporting the Abkhaz separatists after Moscow announced in April that it was establishing official ties with rebels in Abkhazia and South Ossetia, another breakaway region in Georgia.
The Abkhaz order to close the border with Georgia could have an impact on ethnic Georgians in Abkhazia's eastern Gali region.
They often cross into Georgian-controlled territory to trade and visit relatives.
Abkhazia broke away from Georgia in the early 1990s after a war that killed several thousand people.