Rayshard Brooks, 27, was shot dead as he fled officers in a restaurant car park in Atlanta late on Friday.
The killing sparked protests. A top House Democrat said Mr Brooks' actions "did not call for lethal force" - echoing a statement by Atlanta's mayor.
The city's police chief has quit and the officer involved has been fired.
Atlanta is one of many US cities where the killing of George Floyd by Minneapolis police has sparked protests against racism and police brutality.
Mr Floyd, also an African American, died after a white officer knelt on his neck for almost nine minutes late last month.
What happened on Friday night?
The Georgia Bureau of Investigations (GBI) said the officers were called to a Wendy's fast-food restaurant because Mr Brooks had fallen asleep in his car, which was blocking the drive-through lane.
Body camera footage released by the police department shows the officers administering a sobriety test and then a breathalyser test with Mr Brooks' permission.
The two officers then try to handcuff him - at which point their body cams fall off. In security camera video, they can be seen struggling with Mr Brooks on the ground.
He grabs an officer's Taser and breaks free from the officers, running away. As he is chased by the officers, Mr Brooks is seen turning around and pointing the Taser at them, before continuing to run and then being shot.
What has the reaction been?
On Saturday protests were held at the scene and blocked a major road nearby. Crowds set fire to the restaurant and police used tear gas.
Lawyers representing Mr Brooks' family said the officer had no right to use deadly force, saying the Taser which Mr Brooks had grabbed was a non-lethal weapon.
"You can't shoot somebody unless they are pointing a gun at you," attorney Chris Stewart said.