Israel’s Prime Minister Benjamin Netanyahu said every Hamas member was “a dead man” after the first meeting of his country’s emergency government.
Alongside him, opposition figure Benny Gantz said it was “a time for war”.
But US President Joe Biden said he had spoken to Mr Netanyahu and made it clear that Israel must “operate by the rules of war”.
The death toll in Israel has reached 1,200. More than 1,100 people have been killed by Israeli air strikes on Gaza.
Mr Biden said he understood the anger and frustration of the Israeli people but urged Israel to adhere to the principles of the Geneva conventions. He also warned Iran – which has welcomed the Hamas attack – to “be careful”.
Earlier on Wednesday Mr Netanyahu and Mr Gantz agreed to put aside a bitter political rivalry that had escalated into widespread protests.
Mr Gantz told Israeli citizens that the newly-formed government was “united” and ready to “wipe this thing called Hamas off the face of the Earth”.
Alongside Mr Netanyahu and Mr Gantz, the centrist National Unity Party leader and a former defence minister, the new temporary cabinet would also include Defence Minister Yoav Gallant.
The country’s main opposition leader, Yair Lapid, has not joined the alliance. However, Mr Netanyahu and Mr Gantz said in a joint statement that a seat would be reserved for him in the war cabinet.
“During the war period, no bills or government decisions will be promoted that do not concern the conduct of the war,” said a statement.
“All senior appointments will be automatically extended during the war period.”
The emergency government will give broader national consensus to military action. It also brings to the war cabinet two voices who are experts in military strategy. Both Mr Gantz and also Gadi Eisenkot, who joins as an observer, are former Israeli military chiefs of staff.
The announcement of the new cabinet came in the wake of savage attacks by Hamas militants from the Gaza Strip.
Hamas is known to have taken up to 150 people hostage during the surprise attack on Saturday, including elderly people and children.
On Wednesday Israel’s military said thousands of troops were ready near the border with Gaza for a potential land invasion.
As well as attacks by Hamas, Israel has also exchanged fire with the Hezbollah militia in Lebanon and forces in Syria.
Earlier this week, Mr Gallant announced a retaliatory siege of the Palestinian territory – meaning all energy, water and food supplies have stopped.
The formation of a unity government in Israel has been welcomed by ministers including Itamar Ben Gvir, who handles national security.
“Congratulations on the unity, now we have to win,” he wrote on X (formerly Twitter).
The announcement that Mr Netanyahu and Mr Gantz would work together comes after months of protests against attempts by the prime minister and his government to push through controversial judicial reforms.
The demonstrators were supported by Mr Netanyahu’s political rivals, as well as former top officials in Israel’s military, intelligence and security services, former chief justices, and prominent legal figures and business leaders among others.
Hundreds of military reservists, including air force pilots crucial to Israel’s defence, had threatened to refuse to report for service – leading to warnings that it could impair Israel’s military capabilities.
Last month, Israel’s Supreme Court convened to hear petitions against one of the government’s legal amendments, which would limit their own powers.