"If he disturbs the Golan Heights, he will have to bear the consequences," Lieutenant General Benny Gantz said in an address at Haifa University and broadcast on Israeli television.
"We cannot and shall not allow the Golan Heights to become a comfort zone for Assad," he said.
He spoke hours after Israeli troops and Syrian forces exchanged fire across the sensitive ceasefire line on the Golan Heights, but the Jewish state denied Syrian claims one of its vehicles had been destroyed.
Syria claimed to have destroyed an Israeli military vehicle it said had crossed the ceasefire line during the incident.
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Israel earlier said it had responded to fire from inside Syria that hit a military patrol in the Golan Heights overnight, damaging a military vehicle.
Gantz said that at no point did the vehicle enter Syrian-controlled territory and suggested that Syria was fabricating a story with an obsolete Israeli vehicle left behind in Lebanon during Israel's 2006 war there against Syrian ally Hezbollah.
The Golan Heights have been tense since the beginning of the conflict in Syria more than two years ago.