Speaking in an impassioned televised address to the nation, Netanyahu also said he would retain the defence ministry portfolio at least for now and work to convince his coalition partners to remain in the government.
"To go to elections now would be irresponsible," Netanyahu said. "The security of the country is above political considerations."
Education Minister Naftali Bennett's far-right Jewish Home party has threatened to withdraw from the coalition if he is not named defence minister.
Netanyahu spoke of his military experience in his address and said he could not publicly explain the sensitive security moves currently being taken by Israel.
The longtime premier made the appearance as he came under pressure over a controversial ceasefire deal last week that ended the worst escalation between Israel and Palestinian militants in Gaza since a 2014 war.
On Wednesday, Defence Minister Avigdor Lieberman resigned over the ceasefire, throwing the government into crisis.
After Lieberman's withdrawal along with his Yisrael Beitenu party, Netanyahu's government was left clinging to a one-seat majority in the 120-seat parliament.
Netanyahu spoke after a crucial meeting with Finance Minister Moshe Kahlon on resolving the coalition crisis.
Kahlon's office said earlier that his meeting with Netanyahu ended without a conclusion and they would meet again later in the week.
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