The soldier -- who last week shot the Palestinian in the head while he was lying prone on the ground -- has been arrested and strongly condemned by top military officials.
But far-right politicians and protesters have rallied to his cause, criticised the military's response and demonstrated for his release.
Following the soldier's arrest, posters were distributed calling for Eisenkot's resignation.
The chief of staff, seen by some Israelis as a voice of moderation amid a wave of violence that erupted in October, said in the letter that soldiers must always behave professionally.
"We will not hesitate to hold accountable soldiers and officers who do not respect operational and moral criteria that guide us in our actions."
The 19-year-old soldier who shot the Palestinian appeared in a military court on Tuesday as several hundred of his supporters protested outside.
Prosecutors were seeking to extend his remand in the case, which has gripped the country and sparked political tensions, and the judge decided that he be kept in custody until Thursday.
Video of Thursday's killing in the flashpoint West Bank city of Hebron spread widely online and threatened to further inflame Israeli-Palestinian tensions.
It showed a 21-year-old Palestinian, who along with another man had allegedly stabbed a soldier minutes earlier, lying on the ground, apparently after being shot.
The soldier then shoots him again, in the head, without any apparent provocation.
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