Mohammed Sinokrot, 16, was wounded by gunfire in the Wadi Joz neighbourhood on August 31.
Police said he was shot in the leg with a foam pellet 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.
"This is an assassination." 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 what he described as a "foam-coated pellet".
Such pellets are sometimes used by police forces as alternatives to other non-lethal projectiles such as rubber bullets.
Despite being shot, 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.
After initial treatment in Maqased hospital in east Jerusalem, he was transferred to Hadassah Ein Kerem in west Jerusalem where he was pronounced dead on Sunday.
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.
There have been almost nightly clashes in east Jerusalem since July 2 when a Palestinian teenager was kidnapped and burnt alive by Jewish extremists in a revenge killing for the abduction and murder of three Israel teens in the West Bank in June.
Since then, police have arrested more than 600 Palestinians on charges of disturbing public order in east Jerusalem, which was captured by Israel in the 1967 Six Day War and subsequently annexed in a move not recognized by the international community.
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