Israel is already conducting counter-terrorist operations against Hezbollah. These range from artillery and air strikes across the border to targeted killings deeper inside Lebanon, albeit the latter have been more limited in recent times. But then the latter appears to be picking up speed lately:
“Israel is turning from defending to pursuing Hezbollah, we will reach wherever the organization operates, in Beirut, Damascus and in more distant places,” Gallant added.
(And then explosions in Syria, on both Hezbollah and Iran's own high-value targets.)
I suppose your real question is when does it become justified to pursue the total destruction of Hezbollah like it's been the case with Hamas recently, i.e. transition from "mowing the grass" to whatever euphemism is used for the high[er]-intensity phases of the conflict. (Someone proposed "taking out the trash"; I myself suggested here "ploughing the field".) The main issue with that is not justification but feasibility. Israel already tried that in 2006 and before that engaged in a long campaign 1985-2000. For reasons that are complicated, the IDF eventually pulled back short of achieving this ultimate goal of eradicating Hezbollah. Realistically, Israel would have to entirely occupy Lebanon for years or even decades to entirely suppress Hezbollah from re-emerging there. It might find that the trade-offs are not worthwhile.
I suppose what might be in the cards with the current Israeli government in place is re-occupying a wider buffer zone inside Lebanon. That would prevent some of the shorter-ranged attacks from reaching Israel proper. But with Hezbollah even supplied with [Iranian] top-attack ATGMs these days, it's somewhat of a tricky proposition what option would best limit future Israeli casualties, overall.
A second senior Biden administration official said there are elements inside the Israeli government and military in favor of an incursion. There’s “a growing group that says: ‘Hey, let’s just take a shot. Let’s just do it,’” the senior official said, adding that any incursion could lead to a “major, major escalation that we don’t even know the proportions of.” [...]
In recent days Israel has been stepping up its air campaign, bombing deeper into Lebanon. Strikes last week came within 27 miles of the capital Beirut, the farthest into Lebanese territory from the border since the violence started immediately following Hamas’ October 7 massacres in Israel.
On Monday, Israel struck the Hezbollah stronghold Baalbek in the northeast of the country.
“There are fears this will grow to an expansive air campaign reaching much further north into populated areas of Lebanon and eventually grow to a ground component as well,” another person familiar with the US intelligence said.
Israel’s top general visited the northern border Tuesday and said that Hezbollah “must pay a heavy price” for its actions since October 7.
“It’s clear that the first thing we need to do is push back the enemy. Then, create a very strong barrier,” said Lt. Gen. Herzi Halevi.
Likewise, there's a bit of a competition among Israeli politicians on getting hawkish on Lebanon:
Some Israeli leaders, such as Benny Gantz, a member of Prime Minister Netanyahu’s war cabinet, remarked in February 2024 that Israel could strike the Lebanese military as part of a broader war. “It is important that we be clear—the one responsible for the fire from Lebanon is not only Hezbollah or the terrorist elements that carry it out, but also the government of Lebanon and the Lebanese state that allows the shooting from its territory,” Gantz said. He also added: “There is no target or military infrastructure in the area of the north and Lebanon that is not in our sights.”
Right-wing Israeli politicians who are not presently in government (due to spats with Netanyahu) like Avigdor Liberman have openly called for a re-occupation of Southern Lebanon:
The former defense minister says Israel’s military must “close off” a swath of southern Lebanon and push the terror group north of the Litani river, even if it means 50 years of occupation.
“It can’t be that there are entire towns where close to half of the buildings were simply destroyed,” he says during his Yisrael Beytenu party’s weekly faction meeting, referring to northern Israel where structures have suffered missile damage.
“We will not annex anything, and we will not build settlements, but we will release the territory only when there is a government in Beirut that knows how to exercise its sovereignty.”
“Everything between the Litani and Israel must be under the control of the IDF,” he says, comparing it to the post World War II military occupation of Germany. “If Lebanon won’t pay in territory we haven’t done anything,” he declares.
Alas, while Hamas could be sealed off its supply pretty easily, Lebanon borders Syria, which is far from being as cooperating as Egypt is, with Israel.
Because of Hezbollah’s close relationship with Iran, it is likely that Tehran would resupply Hezbollah quickly if it used this arsenal in a conflict with Israel. This resupply is easier than in the past, as Iran’s presence in Syria expanded considerably after Tehran came to the rescue of the Syrian regime when civil war broke out after 2011, creating a land bridge that enables weapons to go from Iraq to Syria to Hezbollah in Lebanon. This stands in sharp contrast to Hamas, where weapons and people must be smuggled via tunnels from Gaza.
And there's Hezbollah in Syria as well, and Israel has been bombing them too, with tacit cooperation of Russia. (And the Russian game is complicated too: on one hand they are competing with Iran for influence in Syria. On the other, they somewhat need Iran's cooperation against Ukraine, even if on a limited scale: Shaheds-cum-Gerans are a fact, and there's speculation about a short-range missiles deal etc.)
Anyhow, repeated invasions of Lebanon have been part of Israel's "100-year war"--to use the words of IDF commander and later politician Raphael Eitan. (That analogy is not totally out of fashion with others.) And so we might not have seen the last of these ground operations.
And to answer the core of the question, the issue is thus not mainly a legal, but political one. The will of whoever runs Israel matters, and to some extend the will of the main foreign supporter of Israel--the US. One might guess that a different US president, more attuned to the Israeli right-wing vision, and who's shown a lower sensitivity to civilian casualties in the 'shithole countries', or just for taking things up a notch with Iran may make a more intense anti-Hezbollah operation more likely to have US support too. (There's also a "bomb Iran" chorus in the legislative circles in the US.)
BTW, my armchair strategizing says that Hezbollah cannot afford to go 'all out' with their missile arsenal against Israel unless either they're threatened with losing it to a ground invasion (a 'use it or lose it' situation) or Iran itself gets seriously bombed, e.g. if Trump greenlights the taking out of Iran's nuclear installations. At least publicly, even the Biden administration is being strategically ambiguous about that, or even hinting at supporting the Israelis in whatever they'd do in that (Iranian nuclear program) regard. OTOH the present US administration has said they are against a war in Lebanon.