The incident on Thursday led to the soldier's arrest and strong condemnation from military officials, but some rightwing politicians and their supporters have reacted angrily to the response.
Netanyahu on Thursday joined condemnations of the soldier's behaviour but appeared to backpedal slightly Sunday in remarks at the start of a cabinet meeting.
"Any challenge to the morality of the IDF (Israel Defence Forces) is outrageous and unacceptable," he said.
"I am certain that in all cases, as in the current one, the inquiry takes into account all conditions. We must all support the IDF chief of staff, the IDF and our soldiers, who safeguard our security."
Netanyahu leads one of the most rightwing governments in Israeli history but holds only a one-seat majority in parliament.
Video from Thursday's incident in Hebron in the occupied West Bank spread widely online and threatened to further inflame tensions amid a wave of violence that erupted in October.
The soldier then shoots him again, in the head, without any apparent provocation. The Palestinian, Abdul Fatah al-Sharif, was killed.
Sarit Michaeli, a spokeswoman for Israeli rights group B'Tselem, which posted the video, called it an "execution".
The UN's special coordinator for the Middle East peace process, Nickolay Mladenov, called it "a gruesome, immoral and unjust act".
The soldier was detained, and Defence Minister Moshe Yaalon vowed that the incident would be treated with "utmost severity".
Bennett questioned whether the soldier thought the Palestinian had explosives on him, though there were reports that he had already been checked for a suicide belt before the shooting.
"The soldier is not a murderer," Bennett wrote on his Facebook page.
"Have we lost our minds? We are at war, a war against brutal terrorism.
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