Israel's military has also assigned bodyguards to the three judges who found the 20-year-old French-Israeli soldier guilty yesterday, reports said.
Sgt Elor Azaria was convicted after a trial in a military court that began in May and which deeply divided the country.
It has led to harsh criticism of the verdict by far-right activists, while right-wing politicians -- including Prime Minister Benjamin Netanyahu -- have called for the soldier to be pardoned.
According to Israeli news site Ynet, the posts by a 54-year-old man and 22-year-old woman included death threats against the head of the three-judge panel in the case, Colonel Maya Heller.
"Police units are continuing to monitor social media and respond to threats and incitement to violence connected to the Azaria court decision," a statement said.
Military and justice ministry officials declined to comment on the increased security for Heller as well as the two other judges, Colonel Carmel Wahabi and Lieutenant Colonel Yaron Sitbon.
In a sign of the tensions surrounding the case, dozens of protesters scuffled with police Wednesday as they gathered outside Israel's military headquarters in Tel Aviv, where the verdict was announced.
Military chief of staff Gadi Eisenkot was reportedly targeted with threats, with some supporters of the soldier suggesting he would join Rabin -- a reference to the 1995 assassination of former prime minister Yitzhak Rabin by a Jewish extremist.
The case burst into public view when a video of the March 24 shooting in Hebron in the occupied West Bank emerged and spread widely online.
Azaria then shoots him again in the head without any apparent provocation.
Heller spent more than two and a half hours reading out the decision yesterday, sharply criticising the arguments of Azaria's lawyers.
The judges ruled there was no reason for Azaria to open fire since the Palestinian was posing no threat. He faces up to 20 years in prison and will be sentenced at a later date.
A poll by pro-Netanyahu newspaper Israel Hayom found that around 70 per cent of Israelis favour a pardon for Azaria.
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