"I am announcing that the Habayit Hayehudi ministers will oppose a Hamas ceasefire that does not completely halt the launching of firebomb kites and the rearming of Hamas", Bennett said, reiterating that "a ceasefire without preventing the intensification and stopping the kite terrorism would be a mistake we cannot accept".
Since April, groups of Gazans have been launching kites, balloons, and even condoms attached to incendiary devices and causing fires, burning almost 2,500 acres of fields, parks and nature reserves in southern Israel.
The primary target of the IAF strikes Saturday was the Hamas battalion headquarters in Beit Lahia, in the north of the Strip, the army said.
The low-tech incendiary devices have become a popular way for Palestinian rioters along the Gaza border to inflict costly destruction on Israeli land without putting themselves in danger.
Smoke rises from Israeli agricultural fields near the Gaza Strip border, after being set on fire by a molotov cocktail kite flown over by Palestinians as they protest by the border fence on May 14, 2018.
The Israeli military said Hamas was responsible for a steady escalation of violence along the frontier.
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Last week, Israel closed the crossing to commercial trade, a move Hamas denounced as a "crime against humanity".
It appeared to be the first case of Gaza militants using birds to attack Israel and authorities said they were considering filing a complaint - under the appropriate worldwide treaties - about the use of animals for militant activities.
During the cabinet meeting, Israeli Defense Minister Avigdor Liberman warned Israel has "no intention of tolerating this - not rockets, not kites, not drones - nothing".
The government is under pressure from local communities to show zero tolerance to this new threat, and Netanyahu told local leaders he had instructed the military to halt it completely.
Amid a shaky Egyptian-brokered ceasefire reached on Sunday, Israeli Prime Minister Benjamin Netanyahu hs affirmed that "there is no such thing as a ceasefire for us that excludes the flaming kites and balloons".
Gaza's only other functioning border crossing, Rafah, is with Egypt.
The IDF said Iron Dome intercepted 20 projectiles in total that were headed for residential areas.
More than 130 Palestinians have been killed by Israeli forces in the protests.