Mohammed Sinokrot, 16, was wounded by gunfire in the Wadi Joz neighbourhood where he lived, on August 31.
News of his death sparked widespread clashes near his home and throughout the Israeli-annexed east of the city.
Police said masked Palestinians threw stones and firecrackers at officers in Wadi Joz and the Issawiya and A-Tur districts of east Jerusalem and stoned civilian traffic.
They also threw petrol bombs at a petrol station on the edge of the east Jerusalem Israeli settlement of French Hill, setting a pump on fire.
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No serious injuries were reported on either side.
Police said Sinokrot was shot in the leg with a sponge grenade -- intended to be a non-lethal form of crowd control -- while rioting, but his family said he was shot in the head on his way to the mosque.
"There were no clashes in the area, he went for night prayers at the mosque and was bringing bread back home," his uncle Muhtadi Sinokrot told AFP.
"Whatever they call the bullet -- it caused Mohammed's death, it broke his skull and caused internal bleeding," he said.
Police spokesman Micky Rosenfeld told AFP the border police had identified Sinokrot as being "involved in rioting" and he was "shot in the leg" with the projectile, which has a foam-rubber nose atop a high-density plastic body and is fired from a grenade launcher.
Despite being hit, Sinokrot attempted to flee but he "fell and then was transferred to Maqased hospital," Rosenfeld said.
Rosenfeld said Sinokrot must have hit his head when he fell. He said the justice ministry's internal affairs unit was looking into the incident, in what he described as "standard procedure" for such cases.
A hospital spokeswoman told AFP his body was being transferred to the Abu Kabir forensic institute in Tel Aviv for an autopsy.
But Sami Sinokrot, another uncle of the teenager, said the family do not want a post-mortem.
"We don't want an autopsy, we know why he was killed," he told AFP, saying police had shot him "point blank" at a time when there were no clashes in the area.