The violence showed no sign of abating as Israeli Prime Minister Benjamin Netanyahu on Friday vowed to "extract a very heavy price" from Hezbollah for the killing of the four soldiers.
Paris urged Washington to put pressure on Israel to stop hostilities in Lebanon, where the intensifying violence strained an interim deal between the United States and Iran halting the broader Middle East war.
The deal requires the United States, Iran, and their allies to declare an immediate and permanent termination of military operations on all fronts, including Lebanon.
Violence abated significantly earlier this week, but has since picked up.
Israel's leadership has promised to continue the occupation of Lebanon in defiance of the US-Iran agreement, which calls for Lebanon's sovereignty to be respected.
In a statement on Friday, Israeli Defence Minister Israel Katz said Israeli forces would remain in southern Lebanon "from the Mediterranean coast to the heights of Beaufort".
In an interview on Israeli television, Katz said the most important military goal was to hold territory.
He said the Israeli military was destroying villages in the areas it occupied and would never allow people to return to their homes.
Lebanon's health ministry said 18 people had been killed and 33 wounded in heavy air strikes in 11 towns since midnight, and that bombardment was preventing rescue and evacuation efforts.
Israel said it carried out strikes targeting what it described as Hezbollah operatives and infrastructure across several areas of the south, saying these were in response to repeated ceasefire violations by the Iran-backed group.
Hezbollah denied it had violated the ceasefire, and accused Israel of repeatedly violating the ceasefire terms, including the US-Iran agreement.
Heavy fighting overnight was concentrated in an area north of the Litani River known as Ali al-Taher hill - high ground strategically important to Hezbollah where Israeli forces had sought to advance, a senior Lebanese security source said.
Hezbollah said its fighters ambushed an Israeli force advancing near the hill, destroying three Merkava tanks with guided missiles and targeting troops with rocket and artillery fire.
Hezbollah said it later attacked Israeli forces that had sought to enter the area to retrieve casualties.
The Israeli military said four soldiers had been killed in an incident in Lebanon, but without giving further details.
Lebanon was sucked into the regional war when Hezbollah opened fire at Israel on March 2, prompting Israel to launch a major offensive against the group and invade the south.
Israel has rejected calls to withdraw troops from southern Lebanon, where its forces are occupying a self-declared security zone.
Israel says this aims to shield northern Israel from Hezbollah attack.
Hezbollah has continued to launch attacks on Israeli positions in the south, including with explosive drones that have killed and injured troops.
