A defence lawyer said on Wednesday the delay was granted so that experts can review a tape of the incident.
Army Captain Rogelio Maynulet, a 29-year-old from Chicago, denies the charges of murder and dereliction of duty in the 21 May death of the driver.
A military court in Hanau, Germany granted a delay until 6 October, which will determine whether Maynulet will be court-martialed, said his lawyer, Captain Will Helixon.
The article 32 hearing - the military equivalent of a US grand jury investigation - originally was scheduled to conclude last Friday.
Maynulet was leading his 1st Armoured Division tank company on a patrol when it came across a BMW saloon believed to be carrying al-Sadr militiamen and a chase ensued. US soldiers fired at the vehicle, wounding both the driver and passenger.
Video details unclear
The recording by a US military drone aircraft shows the incident but it is unclear how much detail it provides.
Actions of some US soldiers have
been a PR nightmare for officials
Reporters were ordered out of the courtroom when it was shown last week allegedly out of concern that it could give away information about the Army's technical capabilities.
The defence has asked that the video be declassified and made available to the public. A military neurosurgeon last week testified behind closed doors about its contents, and Helixon has called two neurosurgeons of his own to analyse the video.
The delay was granted so that those witnesses - one military and one civilian - can arrange a viewing and so that the civilian expert can try to get the necessary clearance.
Witnesses at the hearing have testified that the victim was critically injured before Maynulet came across him.
A fellow officer said in court last week that Maynulet told him a medic had said the man, whose head had been blown open, was beyond hope and that he had shot him out of compassion.
Neither the prosecution nor the defence would comment on the contents of the tape, but it could help determine whether the driver could still have been saved at the time of the alleged killing.