Israel's Foreign Ministry said in a statement that Katz had asked German Foreign Minister Heiko Maas to convey the following message to Beirut: "If you do not stop Hezbollah's activities against Israel, then all of Lebanon will suffer damage and be severely affected."
Katz said Israel did not want an escalation in violence but that his country "is prepared to continue to react harshly to any attack and sees the state of Lebanon as the only responsible party."
Germany is acting as a go-between and the Foreign Ministry in Berlin said Maas had telephoned counterparts in Lebanon and in Israel in recent days. Maas told Beirut that a moderating influence was needed on Hezbollah.
Israel's Katz has called for German sanctions against the militant group and demanded that Berlin classify Hezbollah as a terrorist organization.
Calm had returned to the Lebanese-Israeli border on Monday, following Sunday's burst of violence.
According to United Nations sources, members of the United Nations Interim Forces in Southern Lebanon (UNIFIL) were patrolling the land around Maroun al-Ras, which was hit by more than 40 incendiary and cluster devices shortly after Hezbollah missiles damaged an Israeli military position and ambulance.
Israel said no one had been hurt on their side of the border. There was no word on Lebanese victims.
Lebanese local media said Sunday's missile strike was a response to an Israeli attack which targeted Hezbollah militants in Syria last month. Israel said the attack thwarted an Iranian attempt to hit Israeli targets using "killer drones."
A source with close links to pro-Iranian Hezbollah said that there would be another attack in response to a suspected Israeli drone attack on a Hezbollah stronghold in Beirut's southern suburbs on August 25.
Israel has said Iranian officers are working with Hezbollah in Lebanon to advance an effort to convert rockets into precision-guided missiles.
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