The developments came as US Secretary of State Marco Rubio, speaking from Jerusalem, called on Lebanon to disarm Hezbollah following its recent war with Washington's closest regional ally Israel.
Lebanon's National News Agency (NNA), meanwhile, reported Israeli forces opened fire towards the southern border town of Hula "after residents entered", killing a woman.
The violence followed unrest this week over the government's decision to block Iranian flights, which saw a UN peacekeeper convoy attacked during a protest.
"Israel must fully withdraw on February 18, it has no excuse," Hezbollah's Naim Qassem said in a televised address.
"It is the responsibility of the Lebanese state" to exert every effort "to make Israel withdraw", he added.
A fragile ceasefire between Israel and the Iran-backed militant group has been in effect since November 27 after more than a year of hostilities including two months of all-out war.
Under the deal, Lebanon's military was to deploy in the south alongside United Nations peacekeepers as the Israeli army withdrew over a 60-day period that was later extended to February 18.
Hezbollah was to pull back north of the Litani River -- about 30 kilometres (20 miles) from the border -- and dismantle any remaining military infrastructure in the south.
Both sides have accused the other of violations.
NNA also reported on Sunday three Israeli air strikes in the eastern Bekaa Valley.
The Israeli military said it conducted air strikes targeting Hezbollah military sites storing weapons including rocket launchers in Lebanon, without specifying where.
- 'Must be disarmed' -
During a joint address with Israeli Prime Minister Benjamin Netanyahu, top US diplomat Rubio said that "in the case of Lebanon, our goals are aligned... A strong Lebanese state that can take on and disarm Hezbollah".
Netanyahu said Israel would do what it has to in order to "enforce" the ceasefire.
"Hezbollah must be disarmed. And Israel would prefer that the Lebanese army do that job, but no one should doubt that Israel will do what it has to do to enforce the understandings of the ceasefire and defend our security," Netanyahu said.
Hezbollah was left weakened by the war, which saw a slew of senior commanders and even its longtime chief Hassan Nasrallah killed in Israeli strikes.
Qassem called for broad participation in Nasrallah's funeral, set for next Sunday, as a show of the group's strength.
Protests erupted this week when authorities blocked Iranian planes from landing in Beirut, and a UN convoy was attacked near Beirut airport during a protest involving Hezbollah supporters, wounding two peacekeepers.
Qassem said that the prime minister's office had been informed "that Israel will strike the Beirut airport runway if the Iranian plane lands".
Israel's military warned this week that Iran's Quds Force and Hezbollah were using civilian flights to smuggle money for re-arming the Lebanese group.
Israel has previously accused Hezbollah of using Beirut's airport to transport Iranian weapons, allegations the group and the Lebanese authorities deny.
- 'Gravely mistaken' -
"The prime minister decided to prevent it under the banner of aviation and civilian safety... The problem is that this is implementing an Israeli order," Qassem added.
A Lebanese source told AFP on Saturday that Lebanon had denied permission for Iranian flights to land twice this week, after the United States warned Israel might strike the airport.
Hezbollah earlier Sunday urged the government to reverse the decision.
Iran's state news agency IRNA quoted the head of its civil aviation organisation Hossein Pourfarzaneh as saying the body was "following up on this issue daily" and "waiting to see what will happen on February 18".
Hezbollah lost a supply route when Islamist-led rebels in December ousted ally Bashar al-Assad in neighbouring Syria.
Rubio said that "while the fall of Assad is certainly promising", Washington would be watching Syria "very carefully".
Netanyahu warned "Israel will act to prevent any threat from emerging near our border in southwest Syria".
"If any force in Syria today believes that Israel will permit other hostile forces to use Syria as a base of operations against us, they are gravely mistaken," he added.
Israel conducted hundreds of strikes in Syria after war broke out there in 2011, mainly targeting Assad government forces and pro-Iran groups including Hezbollah.
It also conducted strikes after Assad's fall, and Israeli troops have entered the UN-patrolled buffer zone separating Israeli and Syrian forces in the Israeli-annexed Golan Heights.
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