Al Jazeera's Scott Heidler, reporting from Jerusalem on Sunday, said that the request to revoke the credentials cover all the networks journalists in both the Arabic and English channels.
Israel has long presented itself as a bastion of press freedom in a region where government restrictions on the media are commonplace, and has pointed to Al Jazeera's presence in Jerusalem as proof of the nation's liberal values.
Mr Netanyahu tweeted his congratulations to Mr Kara, "who on my instructions took concrete steps to end Al Jazeera's incitement".
Ayoub Kara alleged that the channel supported "terrorism", and said both its Arabic and English-language channels would be taken off air.
It also denied the charges its coverage of al-Aqsa Mosque unrest was unprofessional.
The latter four Arab states have included shutting down Al-Jazeera in a list of demands made to Qatar, which they accuse of supporting extremists.
Kara wants the Knesset, Israel's unicameral parliament, to consider his request in the next session.
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The broadcaster "will follow up the subject through appropriate legal and judicial procedures", he added.
Prime Minister Binyamin Netanyahu has been threatening to shut Al-Jazeera's operations in the country, accusing the network of inciting violence against Israel.
"Changing the law in order to shut down a media organization for political reasons is a slippery slope", the executive secretary of Israel's Foreign Press Association, Glenys Sugarman, told the news agency Reuters. Israeli authorities would also seek to limit access by the state's Arab citizens to the station's broadcasts in Arabic, the communications ministry statement said without elaborating.
"And when we see that all these countries have determined as fact that Al-Jazeera is a tool of the Islamic State, Hamas, Hezbollah and Iran, and we are the only one who have not determined that, then something delusional is happening here", he said. "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.
"Our coverage of events in the palestinian territories is professional and objective, and the Israelis have acknowledged a few times because we are keen to present the views and their opposites", he said.
Palestinians refused to enter the site for almost two weeks over the new security measures imposed after three Israeli Arabs shot dead the policemen nearby.