"My stance has not changed: Pardon for Elor Azaria," Netanyahu posted to Facebook.
Azaria, an Israeli Defense Forces medic, was convicted of manslaughter for shooting Palestinian terrorist Abdel Fattah al-Sharif after he was subdued in the West Bank city of Hebron. The shooting was captured on video by a Palestinian human rights activist.
Azaria, now 20, began serving his 14-month sentence in August.
Netanyahu has previously called for Azaria to be pardoned, even after the military court rejected his appeal.
"The aforementioned letter has not been received ... and when it does it will be answered as procedure dictates," Rivlin's office said. "Pardon requests are accepted only from the person himself [facing imprisonment], his representatives or immediate family members."
The presidential statement added that a pardon can only be filed within six months from the day the president has given a decision on the request "unless the circumstances of the request undergo a significant change."
Last week, Rivlin announced his refusal to pardon Azaria based on "all of the material and professional opinions that had been presented to him."
The president's statement noted that military chief Lt. Gen. Gadi Eisenkot reduced Azaria's original sentence by four months.
Also Monday, Israel's ruling coalition reached a compromise on the so-called "recommendations law," which prohibits law enforcement officials from saying whether charges should be filed against government officials. The law would apply retroactively to two current corruption investigations against Netanyahu.
The law would give Israel's attorney general the option to follow a police recommendation -- instead of an indictment -- as long as the recommendation is not public.
Nembers of Netanyahu's Likud party formulated the compromise. Yair Lapid, head of the opposition Yesh Atid party, called the bill "The Netanyahu Law."
Lapid, considered to be Netanyahu's main rival when elections are next held, said at the Knesset Monday that "it's a law made for a single person" -- and if it's passed, his party will appeal to Israel's Supreme Court of Justice.