The video threatened to further inflame tensions amid a wave of Israeli-Palestinian violence that erupted in October.
It showed the Palestinian, who along with another man had allegedly stabbed a soldier in the West Bank city of Hebron, lying on the ground, apparently after being shot.
The soldier then shoots him again, in the head, without provocation. The Palestinian, identified as 21-year-old Abdul Fatah al-Sharif, was killed.
The soldier was detained, with Defence Minister Moshe Yaalon vowing that the incident would be treated with "utmost severity".
Prime Minister Benjamin Netanyahu said soldiers were obliged to exercise self-control.
"The (army) expects its soldiers to behave calmly and in accordance with the rules of engagement," he said in a statement.
"What happened in Hebron does not represent the values of the Israel Defence Forces."
Military spokesman Peter Lerner said "all of the soldiers involved will be investigated, as will the orders given."
Palestinian health minister Jawad Awwad called it a "war crime," saying that even prior to the shooting medics on site were not treating the wounded man.
Violence since October has left 200 Palestinians and 28 Israelis dead.
Most of the Palestinians killed were carrying out knife, gun or car-ramming attacks, according to Israeli authorities.
But Israeli forces have also been accused of using excessive force in some cases, charges which they have firmly denied.
In the video, the man -- who appears to be seriously injured -- is lying on the ground while Israeli medics treat the injured soldier.
The video sparked condemnation.
Sarit Michaeli, a spokeswoman for Israeli rights group B'Tselem, which posted the video, called it an "execution".
"It is clear that the Palestinian youth who was one of the two who stabbed the soldier was lying on the ground without posing any risk to security services," she told AFP.
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