Saleh Mohammed Saljuqi, an assistant to the parliamentary speaker, said: "Those responsible for the accident on Monday should be handed over to Afghan legal authorities."
The national assembly passed a nonbinding motion after debating Monday's anti-US riots that paralysed the capital. The unrest started after a US military truck plowed into a line of cars, killing up to five Afghans. The situation deteriorated when several further people were killed by US gunfire.
Meanwhile, the White House press secretary said that George Bush, the US president, spoke to Hamid Karzai, his Afghan counterpart, and pledged an investigation.
He said: "They talked about recent developments and the need to continue with police reform and capacity building. The president expressed sympathy for those killed and injured in Kabul on Monday and pledged a full investigation."
Up to 20 Afghans were killed and more than 160 wounded, mostly from gunshots. The situation has since calmed, but Kabul remains under night curfew.
"Our soldiers used their weapons to defend themselves"
Tom Collins, US army colonel
The unrest was seemingly fuelled by widespread poverty and unemployment.
A US military spokeswoman, Lieutenant Tamara Lawrence, said she had not seen the parliamentary motion and declined to comment, as did a spokesman for Karzai, Khaleeq Ahmed.
It seems unlikely the motion will lead to any action. US military personnel in Afghanistan come under American military jurisdiction, although foreigners working on civilian projects are generally subject to Afghan law.
Another military spokesman, Colonel Tom Collins, said that the driver of the truck was not suspected of any wrongdoing and had not been arrested. He said the truck's brakes are believed to have overheated and failed.
However, he said the military was investigating whether the troops involved in the crash fired their guns into a group of demonstrators or over their heads. He said some of the rioters who were throwing stones at the US troops also had weapons and were firing at them.
"Our soldiers used their weapons to defend themselves," he said. Asked if this meant that they fired into or over the crowd, Collins said: "Our investigation is still looking into this."
The unrest has shaken confidence in authorities' control of the capital - which is patrolled by Nato peacekeepers and is regarded as relatively secure - in comparison to the south of the country, where fighters loyal to the former Taliban government have made gains in recent weeks.