Iyad Halaq, 32 was killed on Saturday in occupied East Jerusalem while he walked to his special needs school.
Israel's police force said officers suspected Mr Halaq was carrying a weapon and that they opened fire when he failed to obey orders to stop. He was later found to have been unarmed.
Israeli Defence Minister Benny Gantz expressed regret over the shooting.
"I am sure this issue will be quickly investigated and conclusions will be drawn and Israeli will be responsible for what happened and be responsible to the world."
Mr Gantz added that Israeli security forced would "make every effort to use necessary force with the intention of reducing casualties as much as possible".
Tensions have risen in recent weeks after Israeli Prime Minister Benjamin Netanyahu said he would press ahead with a plan to annex parts of the occupied West Bank.
The Palestinian Authority has responded by saying it is no longer bound by agreements with Israel and the United States - which backs Israel keeping part of the West Back - including those relating to security.
Iyad Halaq would walk every day from his home in Jerusalem's Wadi al-Joz area to the Old City to go to the Elwyn EI Quds centre, which provides services for children and adults with disabilities.
Mr Halaq's cousin, Dr Hatem Awiwi, said he was on the low-functioning end of the autism spectrum and that he had trouble communicating with people.
"He didn't know what a police officer is, "Dr Awiwi told Israel's Haaretz newspaper. "He saw a stranger and fled, and then they shot him."
An Israeli police statement said units on patrol in the Old City "spotted a suspect with a suspicious object that looked like a pistol".
"They called upon him to stop and began to chase after him on foot. During the chase, officers also opened fire at the suspect, who was neutralised," it added. "No weapon was found at the scene after the area was searched."
An autopsy carried out on Sunday found Mr Halaq was shot twice in the chest.
"The findings increase the suspicion that the policemen committed crimes, and we expect those responsible for the investigation to move it forward and put the policemen on trial," the Halaq familiy's lawyer, Jad Qadmani, said.
The secretary general of the Palestine Liberation Organisation (PLO), Saab Erekat, said it was "crime that will be met with impunity unless the world stops treating Israel as a state above the law".
He drew parallels with the killing of George Floyd in the US, which has sparked widespread protests.
Palestinians and Israelis protested against the killing of Mr Halaq in Jerusalem, Tel Aviv and Jaffa on Saturday and Sunday. Some carried posters saying "JUSTICE FOR IYAD" AND "PALESTINIAN LIVES MATTER".
@ Jackie San