The Israeli army said it hit back by striking "dozens" of Iranian sites in Syria, targeting intelligence, storage and logistics as well as the origin of the rockets.
"It openly calls daily for our destruction, the elimination of Israel from the face of the earth and it practices unmitigated aggression against us", he added.
Israel said it attacked almost all of Iran's military infrastructure in Syria on Thursday after Iranian forces fired rockets at Israeli-held territory for the first time in the most extensive military exchange ever between the two adversaries.
Social media users report that Israeli tanks have fired at positions in Syria, Israeli air sorties have been initiated, and that Syrian artillery is shelling Israeli targets.
Tehran's growing clout in Syria alarms arch foe Israel, which has struck what it describes as Iranian deployments or arms transfers to Hizbollah scores of times during the conflict. The incoming attack set off air raid sirens in the Israeli-controlled Golan, which was captured from Syria in the 1967 war.
The Syrian Observatory for Human Rights reported Wednesday that Israel has attacked targets of the Syrian military on the outskirts of Quneitra in the Golan Heights.
Expectations of a regional flare-up were stoked by president Donald Trump's announcement on Tuesday that he was withdrawing from the Iranian nuclear deal.
DHS's Nielsen Almost Quit After Trump Border Rant
As head of Homeland Security , Nielsen oversees a 20,000-person work force under Immigration and Customs Enforcement. Kelly told Fox News Friday that after Thursday's cabinet meeting, he phoned Nielsen to implore her not to quit.
Israeli Prime Minister Benjamin Netanyahu has traveled to Moscow where he and President Vladimir Putin will discuss military coordination amid new strikes in Syria blamed on Israel. The report also claimed that Assad air defences intercepted tens of Israeli missiles.
"The meetings between us are always important and this one is especially so", Netanyahu said ahead of his departure.
The United States portrayed its rejection of that agreement as a response, in part, to Iran's military interventions in the Middle East, underpinning Netanyahu's tough line towards Tehran. Israel has warned that it will not allow Iran to establish a permanent military presence in Syria.
It was the heaviest Israeli barrage in Syria since the 2011 start of the civil war in which Iranians, allied Shi'ite Muslim militias and Russian troops have deployed in support of President Bashar al-Assad. After an uneventful night, the military on Wednesday called on residents to return to "full civilian routine", meaning studies and excursions would continue as usual, although the shelters would remain open.
According to the Observatory 15 people were killed, among them eight Iranian fighters.
Amos Gilead, a retired senior Israeli defence official, told a security conference in the coastal town of Herzliya that Iran's intentions in Syria meant a wider conflagration may only be a matter of time. "They are determined to do it and we are determined to prevent it".