The website of Israel's Yediot Aharonot newspaper reported the soldier had been arrested.
Footage of the incident, released by the Israeli human rights group B'Tselem, shows the demonstrator with an army officer holding his arm.
A soldier next to him appears to take aim at his leg, a shot is fired and the camera darts briefly away before showing the man lying on the ground.
The protester, identified as Ashraf Abu Rahman, 27, was lightly wounded in the incident which took place on July 7 in Nilin, an occupied West Bank village where regular protests are staged against Israel's barrier.
The army, which said five border guards, three soldiers and two labourers working on the wall were injured during protests in June, announced the inquiry after the video was shown on Palestinian television.
An Israeli army spokeswoman said the film raised several questions and had "missing sequences".
She added that the man went home with a wounded toe and without filing a complaint.
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