"Al Jazeera deplores this action from a state that is called the only democratic state in the Middle East and considers what it has done is risky", an unnamed official with Al Jazeera told the AFP.
Ayoub said that Saudi Arabia and Egypt "have concluded that Al Jazeera incites terrorism and religious extremism".
Mr Kara accused the station of "supporting terrorism" and said cable broadcasters had agreed to his proposal to take its broadcasts off air.
The Israeli government said it was basing its decision on a similar ban by several Sunni Arab states, amid their diplomatic rift with Qatar, which hosts and funds Al Jazeera.
"Israel claims that it is the only democratic State in the Middle East and boast about it, but it follows, by its decision, that of dictatorial countries that do not recognize freedom of expression and freedom of information", stressed the head of Al-Jazeera.
The move follows Israeli Prime Minister Benjamin Netanyahu's call for the station's offices to be shut down, accusing it of inflaming tensions around Al-Aqsa mosque, the Jerusalem holy site where Palestinians staged an nearly two-week protest last month against Israeli security measures installed after a deadly shootout.
The Legal Center for Arab Minority Rights in Israel, also known as Adalah, also challenged the plan, saying that it would be subject to scrutiny before the Supreme Court, adding that "it would fail the test of legality".
Protests erupted at the contentious site after Israel last month installed new security measures including metal detectors, following the shooting dead of the two Israeli policemen by attackers who emerged from the compound.
Saudi Arabia and Jordan have both shut Al Jazeera bureaux this year as part of the ongoing "cold war" playing out in the Gulf, which culminated in the full blockade of Qatar. "If this does not happen because of legal interpretation, I will work to enact the required legislation to expel Al Jazeera from Israel", Netanyahu added. The site saw violent clashes in recent weeks.
"Al-Jazeera had a big role conveying the Palestinian narrative with a high professionalism", said Hazem Qassem, a Hamas spokesman.
The network's offices in the Palestinian territories of Gaza and the West Bank city of Rammallah, would not be affected by the current Israeli move.