The shootout and detention occurred Wednesday evening in the village of Vashlisdzhvari, outside the capital, Tbilisi, ministry spokesman Guram Donadze told The Associated Press. The suspect fled into the woods but was later detained, Donadze said.
Rustavi-2 television showed pictures of a dark-haired man it described as the suspect being hustled into a car by police officers. It said he was wounded and identified him as Vladimir Arutyunov, in his late 20s.
The man lived in an eight-story apartment building with his mother, Rustavi-2 reported, citing neighbors as saying Arutyunov was unemployed. The report could not immediately be confirmed.
The police operation came two days after authorities released a photograph of a man suspected of throwing the grenade, which failed to explode, at a podium where Mr. Bush was speaking May 10 before tens of thousands of people.
President Mikhail Saakashvili was also on the podium when Mr. Bush spoke, raising the prospect that the grenade could have been directed at him.
In Washington, a spokesman for the U.S. Secret Service, Eric Zahren, said the agency is monitoring the investigation by the Georgian authorities. "We were not directly involved and not present" at the arrest, he said.