"We believe the appropriate sentence for the accused should not be less than three years and not more than five years," the prosecutor, Nadav Weisman, said during a court hearing in Tel Aviv.
Elor Azaria, 20, was convicted this month of manslaughter in a military court for the killing of Abdul Fatah al-Sharif, after a trial that deeply divided Israel.
The March 24 shooting in the occupied West Bank city of Hebron was caught on video and spread widely online.
It showed Sharif, 21, lying on the ground, shot along with another Palestinian after stabbing and wounding a soldier, according to the army.
Azaria then shoots him again in the head without any apparent provocation.
Convicting him of manslaughter on January 4 after a months-long trial, a three-judge panel ruled there was no reason for Azaria to open fire since the Palestinian was posing no threat.
Judge Colonel Maya Heller called his testimony "evolving and evasive".
"His motive for shooting was that he felt the terrorist deserved to die," she said.
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Azaria, who also has French nationality, faces up to 20 years in prison.
His family showed no emotion in court on Tuesday.
The prosecutor said Azaria "acted deliberately, he used his weapon to punish, he killed a person, even if it was a terrorist".
Colonel Gay Hazut, who formerly commanded the unit in which Azaria was serving, said he had "committed something serious and should be punished".
"But," he added, "I do not think he should spend 20 years or even 10 years in prison."
The prosecutor also asked for an unspecified suspended term and for Azaria to be demoted from sergeant to private.
Weisman said the time Azaria had spent confined to base should not be deducted from the sentence, but the nine days he spent in jail could.
The case has sparked political tensions.
Prime Minister Benjamin Netanyahu, who heads what is seen as the most right-wing government in Israeli history, has called for him to be pardoned.
Right-wing ministers have defended Azaria despite top army brass condemning his actions in an extraordinary public rift between politicians and the military.