In a video posted on YouTube and subtitled in Arabic, Netanyahu -- speaking in English -- urged Abbas to dismiss Sultan Abu al-Einein.
Late last month, asked by the Palestinian news site Dunya al-Watan about Palestinian leaders taking part in conferences in Israel, Einein said: "If you ask me my position in theory, I would answer that as soon as there is an Israeli, cut his throat."
Referring to a Palestinian taking part in a conference in Israel, he added that "he has my full confidence -- he carried our voice to them (the Israelis).
"It is essential to maintain relations with this society so it is not held hostage by a group of extremist settlers."
Netanyahu in his YouTube video demanded that the adviser be sacked.
"Your adviser, Sultan Abu al-Einein, recently called to slit the throat of every Israeli. Three days later, a Palestinian terrorist turned these words into action when he slit the throat of a 13-year-old beautiful girl, Hallel Yaffa Ariel, as she slept," the Israeli premier said.
"I ask that you fire this adviser because advocating genocide is not consistent with peace."
On June 30, a 19-year-old Palestinian stabbed the US-Israeli teenager at her home in the Kiryat Arba settlement in the south of the occupied West Bank before being shot dead by guards.
"It is our people who are massacred every day by the Israelis under the direct orders of the Israeli government," Einein responded in a statement.
He also denounced "rabbis who call for Palestinians to be killed" and Israeli Justice Minister Ayelet Shaked "who calls for Palestinian women and Palestinian children in their mothers' wombs to be killed".
Netanyahu accused the Palestinian Authority of paying the perpetrators of deadly attacks on Israeli civilians and financially supporting their families.
"This money provides direct incentive to commit terror," he said.
"I ask that you stop paying murderers and instead use this money to fund co-existence education -- teach tolerance, not terror."