Two-year-old Mohammad al-Tamimi suffered a fatal head wound in the June 1 shooting near Ramallah in the occupied West Bank.
His father was hit in the shoulder.
The Palestinian foreign ministry demanded accountability, deeming the incident a crime.
Publishing the results of its investigation, the Israeli military repeated previous assertions that Palestinian gunmen had fired at soldiers guarding a Jewish settlement that night.
An army officer searching the area saw a "suspicious vehicle and fired several times into the air," prompting a soldier who heard those shots to open fire on the Tamimis' car, believing the gunmen were using it for their getaway.
The investigation faulted commanders for miscommunication and "incorrect decision-making," the statement said, adding that the officer who fired in the air would be reprimanded for violating standing orders.
The Palestinian foreign ministry, in a statement, deplored the findings as "the clearest and ugliest form of disregard for, and legalisation of, the shedding of Palestinian blood".
The West Bank, among territories where Palestinians seek statehood, has experienced a surge of violence over the last 15 months.
A report by Israeli rights group Yesh Din based on military data from 2017 to 2021 said that Israeli soldiers were prosecuted in less than 1 per cent of hundreds of complaints filed against them on alleged offences against Palestinians.
