"It is our right to revenge (Samir Kuntar's) assassination, and we will pick the time, place and manner to do so as we deem appropriate," Nasrallah said in a televised address. "This is our right."
Hezbollah accused Israel of carrying out the airstrike that hit a residential building in the suburbs of Damascus, killing Kuntar. The Israeli military has declined to comment on the accusation.
Nasrallah's vow for revenge came hours after weapons fire was exchanged along the Israel-Lebanon border.
The Israeli military responded with artillery fire into southern Lebanon
after rockets were fired at Israel from its northern neighbor, according to a statement from the Israeli Defense Forces.
The U.N. Interim Force in Lebanon said it detected three rockets fired from the general area of Al Hinniyah in southern Lebanon toward Israel on Sunday evening, and that the "IDF retaliated with approximately eight rounds of 120mm mortar fire that impacted near Zibqin in southern Lebanon."
There were no immediate reports of casualties on either side.
Kuntar served 30 years in an Israeli prison for murder before he was released in a prisoner swap between Israel and Hezbollah seven years ago.
Kuntar was convicted of a 1979 attack in which he shot and killed an Israeli man in front of his 4-year-old daughter, then smashed her head against a rock with his rifle butt, tossing her body into the sea. The victim's 2-year-old daughter was accidentally smothered by her mother, who tried to keep her quiet as they hid from the attackers who arrived in Israel from Lebanon by boat.
Kuntar, who was 16 at the time of the attack, was convicted for murdering both children and was sentenced to 542 years in prison. But in 2008, the Israeli government agreed to release him from prison
in a high-profile swap. Kuntar was traded for the remains of two Israeli soldiers captured by Hezbollah in a 2006 cross-border raid.
"It's not an easy decision," Israeli government spokesman Mark Regev told CNN at the time of the swap, adding of Kuntar: "One of the people being released is a brutal murderer, a man that with his own hands killed infants -- a terrible crime, but this is a nasty business."
Kuntar received a hero's welcome upon his return to Lebanon.
In his address Monday, Nasrallah said Kuntar "was aware that he had always been a target for Israel since the first day of his release."