• Hostilities between Israel and Iran-backed Lebanese militant group Hezbollah have escalated dramatically following nearly 12 months of cross-border strikes.
• Lebanon's Health Ministry said Israel's strikes on Monday killed at least 558 people, including 50 children, and injured more than 1,800.
• Hezbollah has continued firing rockets into northern Israel since the attack, most of which have landed in open areas or been intercepted by Israel's air defenses.

Israeli President Isaac Herzog threatened further military action against Hezbollah one day after Israel launched airstrikes across Lebanon, killing more than 550 people in the country's deadliest day in nearly two decades.

It marks a dramatic escalation of hostilities between Israel and Iranian-backed Lebanese militant group Hezbollah following nearly 12 months of strikes since the start of the Israel-Hamas war in Gaza, stoking fears of an all-out regional war.

Thousands of Lebanese residents in the country's south are fleeing their homes amid the bombardments, with many receiving automated text messages and calls telling them to evacuate. Israel's government issued warnings to people in areas of Lebanon where it was targeting Hezbollah.

Asked if Israel will launch a full-scale ground operation in Lebanon, Herzog insisted that his country did not want war.

"Israel is not interested, did not want this war, and is not interested in going to war with Lebanon," Herzog told CNBC's Dan Murphy on Tuesday.

"But Israel has been attacked from Oct. 8 from Lebanon endlessly. And if you look at the situation today, Hezbollah has launched missiles and rockets all over the northern part of Israel. So we will do whatever it takes to bring our citizens back home and enable calm in our cities. That's the situation." 

"We've shown our capabilities, and we have much more on the way, if they will continue," Herzog added.

Hezbollah has continued firing rockets into northern Israel since the attack, most of which have landed in open areas or been intercepted by air defenses. On Saturday, the group launched a salvo of more than 100 missiles into northern Israel, wounding at least five people, Israeli authorities said. 

Hezbollah said the strikes were in response to the previous week's pager and device detonation attack that killed 38 people, including some children, and injured more than 3,000. It says the attacks were carried out by Israel, which has not commented. 

Lebanon's Health Ministry said Israel's strikes on Monday killed at least 558 people, including 50 children, and injured more than 1,800.

Monday's death toll not only marked the deadliest day of fighting between Israel and Hezbollah since their 34-day war in 2006; it also exceeded that from the 2020 Port of Beirut explosion that killed nearly 200 people and destroyed several of the capital city's neighborhoods.

At the opening of the 79th United Nations General Assembly meeting in New York on Tuesday, U.N. Secretary-General Antonio Guterres warned member states of the risk of Lebanon "becoming another Gaza."

"Gaza is a nonstop nightmare that threatens to take the entire region with it. Look no further than Lebanon," Guterres said, adding that all states should be "alarmed by the escalation" between Israel and Hezbollah.

On Monday, President Joe Biden said the U.S. was trying to calm the situation, while the Pentagon said it stood by "Israel's right to self-defense" and announced that additional U.S. troops were being sent to the Middle East. The U.S. has around 40,000 troops in the region.

Tens of thousands of people on both sides of the Israel-Lebanon border have had to leave their homes as a result of the cross-border fire in the months following Oct. 7 attack on Israel by Hamas. Israel's leaders have vowed that evacuated residents of northern Israel will be able to return to their homes. 

"First and foremost, we have to remove the threat from Israel's northern border, and that's what we're trying to do," Herzog said.

Some lawmakers in Israel's government, which is the most right wing in the country's history, have called for the reoccupation of southern Lebanon. Israel occupied Lebanon's south between 1985 and 2000 — a period of bloody sectarian warfare — following a spate of attacks into Israeli territory carried out by Lebanon-based Palestinian militants.

Asked if reoccupying southern Lebanon was part of the end goal for Israel, Herzog replied, "No, it's not."

"The position of the Israeli government has been clear, and I reiterate it all the time, we don't have territorial expectations or any ambitions in Lebanon or elsewhere," the president said.

"We don't have territorial ambitions in Gaza either, but we have the inherent right, which is given to any nation in the world to live in peace and quiet, not to be attacked day in day out by missiles and rockets and terror." 

Israel launched its invasion of Gaza after the Oct. 7 attack by Hamas militants that killed some 1,200 people in Israel and took a further 253 hostages, 116 of whom have been freed. Since then, Israel's assault on the blockaded enclave has killed more than 41,000 people in Gaza, according to Palestinian health authorities.