US officials had said at the time the grenade failed to explode only because of a malfunction and landed within 30m of Bush as he spoke to tens of thousands of people in Tbilisi's Freedom Square.
The Georgian Interior Ministry released a tape made on Thursday showing the suspect, identified only as Arutunyan, confessing.
Asked by an official whether he threw the grenade, Arutunyan, who looked calm, replied: "Yes."
Deputy Health Minister Irakly Giorgobiani said: "Doctors, who talked to (him) yesterday, said that he had confessed that he had thrown a grenade. But they also said that he may not have been in control of himself at the time."
Georgian police detained the man on Wednesday after a special operation in which an anti-terrorism officer was killed and the suspect wounded.
One man was killed in an operation
to arrest the suspect
Giorgobiani told Rustavi-2 TV the suspect's condition was now normal and there was no threat to his life.
Interior Minister Vano Merabishvili told a briefing the investigation was continuing and that a reward of about $83,000 would be split among several people who helped the investigation.
The security scare marred Bush's trip to Georgia, intended to show US support for the West-leaning government that came to power after a democratic revolution 20 months ago.
Officials said the grenade could have caused major injuries in the close-packed crowd, the largest gathering of people on the square since the "Rose Revolution" led by now-President Mikhail Saakashvili.
The US Embassy in Georgia said in a statement on Thursday
that co-operation on the case was "a model" for joint work by
law-enforcement agencies of the two countries.