In March 2016, Azaria, a combat medic stationed in Hebron, shot a neutralized terrorist in the head as he lay wounded on the ground after committing a stabbing attack against the soldier's colleagues.
Azaria claimed he acted out of fear that Sharif might have been wearing an explosive vest but the judges said his motive was "revenge".
"After 18 months of turmoil, it is time to pardon Elor Azaria".
The family said it "opposes all forms of violence, by Palestinians and Israelis alike".
Azaria demanded for reconsideration of his conviction while the prosecutors requested a harsher sentence.
An IDF Appeals Court today, Sunday afternoon, 7 Av, is expected to announce its verdict in the appeal filed on behalf of former IDF soldier Sgt. Elor Azariya.
As Azaria arrived in the court at the Defence Ministry complex in Tel Aviv, his supporters demonstrated outside.
An Israeli military court on Sunday rejected the appeal of Elor Azaria the Hevron soldier who was jailed for 18 months for killing a wounded terrorist who stabbed and severely wounded an IDF soldier.
Azaria was released from military prison to house arrest, as his period of mandatory army service came to an end and he awaited a decision from the military court of appeals.
He is eligible for pardon by either EiZenkot or President Reuven Rivlin.
"My opinion has not changed on the question of granting a pardon to Elor Azaria, as I expressed it after the [January] conviction".
"The court's message is no less important [than the sentence itself]", said Meretz Chairwoman Zahava Galon.
"I have no doubt that the chief of staff will take all of the hard circumstances, and the fact that he (Azaria) was an exemplary soldier, into consideration", Mr Lieberman tweeted.
The March 2016 shooting in the occupied West Bank city of Hebron was recorded on video by a rights group and spread widely online.
The case has deeply divided Israeli public opinion between those who believe he was wrongly tried and those who say the conviction was the right and proper effect of his actions.
Military leaders strongly condemned Azaria's actions.
Other right-wing lawmakers including MK Ofir Akunis (Likud), MK Eitan Cabel (Labor), MK Betzalel Smotrich (Jewish Home) all also called for the soldier to be pardoned.