Defence lawyers announced an appeal and right-wing Israeli ministers immediately called for a pardon, while the Palestinian government and the victim's family condemned the sentence as too lenient.
Judge Maya Heller handed down the sentence a month after Elor Azaria, 21, was found guilty of manslaughter for killing Abdul Fatah al-Sharif in the southern occupied West Bank in March last year.
He also received 18 months in suspended sentences and was demoted from sergeant to private.
Heller said the panel of three judges had taken into account mitigating factors including the "harm suffered by his family" and the fact Azaria was in "hostile territory" when the shooting occurred.
She added, however, that he had not expressed remorse for his actions.
Azaria will begin his sentence on March 5, the court announced.
He has already spent 10 months in detention at an Israeli military base and it was not immediately clear if Tuesday's sentence includes time served.
Azaria and his family didn't react in court as the sentence was passed down.
The March 2016 shooting in the occupied West Bank city of Hebron was caught on video by a rights group and spread widely online.
It showed Sharif, 21, lying wounded on the ground, shot along with another Palestinian after stabbing and wounding a soldier, according to the army.
Azaria then shoots him in the head without any apparent provocation.
Calls for pardon
Azaria says he feared Sharif was wearing an explosive belt and could blow himself up, taking nearby soldiers and onlookers with him, a claim judges rejected.
The case, which came against the backdrop of a wave of Palestinian knife, gun and car-ramming attacks that erupted in October 2015, has stirred controversy.
Some on the right have defended the soldier, a French-Israeli national, but military officials said the trial was important to maintain the institution's reputation.
Azaria entered the packed courtroom to applause from friends.
Dressed in military uniform and smiling broadly, he hugged family members and his girlfriend.
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Outside the court, around 100 right-wing demonstrators protested against the trial.
As the verdict was read out the family sat still but afterwards they sang the Israeli national anthem together.
Defence attorney Ilan Katz told reporters outside the courtroom they would appeal the manslaughter conviction as it was "more severe than the punishment he got".
Right-wing Education Minister Naftali Bennett and Culture Minister Miri Regev led calls for a pardon.
"Israel's security demands he be pardoned," Bennett wrote on Twitter.
"Elor was sent to protect Israelis at the height of a wave of Palestinian terror attacks. He cannot go to jail or we will all pay the price."
Defence Minister Avigdor Lieberman called on people to respect the ruling.
But he said: "You have, on the one hand, an exemplary soldier, and on the other, a terrorist who tried to kill Jews".
There was no immediate reaction from Prime Minister Benjamin Netanyahu, who has previously said he too favours a pardon.
A source close to President Reuven Rivlin, who could hand down the pardon, said no such request had been made as yet.
Verdict a 'farce'
The reaction from Palestinians and rights groups, however, was anger and disappointment.
A spokesman for the Palestinian government told AFP the sentence was a "green light to the occupation army to continue its crimes".
Sharif's father Yusri told journalists at the family home near Hebron in the southern West Bank the sentence was a "farce".
"What does a year and a half mean? Was he an animal to be killed like this, in this barbaric way?"
Magdalena Mughrabi from Amnesty International said Azaria's initial conviction had "offered a glimmer of hope for accountability for unlawful killings by Israeli forces".
"(But) the 18-month sentence for Elor Azaria does not reflect the gravity of the offence."
Sari Bashi, Israel and Palestine advocacy director at Human Rights Watch, called the conviction an "important message about reining in excessive use of force".
She warned, however, that pardoning him would "encourage impunity".
The trial opened last May at a military courtroom in Tel Aviv's Jaffa district but later moved inside to a tightly guarded military complex.