Israel and Hezbollah threatened on Sunday to escalate their cross-border attacks despite a chorus of international calls for both sides to step back from the brink of all-out war.
Prime Minister Benjamin Netanyahu said after intense rocket fire from Lebanon that Israel has dealt "a series of blows on Hezbollah that it could have never imagined".
A defiant Hezbollah deputy chief Naim Qassem said the group was in a "new phase" in its battle against Israel.
Both spoke after attacks on northern Israel sent hundreds of thousands of people to bomb shelters and caused damage in the Haifa area.
"No country can tolerate attacks on its citizens," Netanyahu said nearly a year into the Gaza war sparked by Hamas's October 7 attack on Israel that has also drawn in Iran-backed groups across the region, including Hezbollah.
Defence Minister Yoav Gallant said military actions "will continue until we reach a point where we may ensure the safe return of Israel's northern communities to their homes".
"This is our goal, this is our mission, and we will employ the means necessary to achieve it."
Army chief Lieutenant General Herzi Halevi in a video statement vowed to "hit anyone who threatens" Israelis.
Israel's key ally the United States said military escalation is not in Israel's "best interest", with President Joe Biden saying Washington was doing everything possible to prevent a wider conflagration.
Biden said his administration was "going to do everything we can to keep a wider war from breaking out. And we're still pushing hard."
Ahead of the annual General Assembly, UN chief Antonio Guterres warned of the risk of Lebanon becoming "another Gaza" and said it was "clear that both sides are not interested in a ceasefire" in the Gaza war.
Hezbollah rocket fire reached Kiryat Bialik near north Israel's largest city Haifa, leaving a building in flames, another pockmarked with shrapnel and vehicles incinerated.
"This is not pleasant. This is war," said resident Sharon Hacmishvili.
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