In an interview with Israel's Channel 10, the minister said Israel's assassination operations must continue in the south "as of tonight", while pointing an accusing finger at Hamas leaders who he said were "responsible for the escalation and deterioration in the security situation".
With limited options at its disposal, and a failure so far of the protests to significantly ease the blockade, Hamas appears to be gambling that limited rocket fire might somehow shake up the situation.
The Israeli military said most of the projectiles fired Tuesday were intercepted, but three soldiers were wounded.
"Quiet will be met with quiet and hostility and violence will be met with appropriate measures", he added.
Egypt intervened on Tuesday to defuse tension after Israel launched the most devastating attacks on Gaza since the bloody conflict in 2014. He said the factions, including Islamic Jihad, would agree to the cease-fire as long as Israel did the same.
"The Security Council should be outraged and respond to this latest bout of violence directed at innocent Israeli civilians", U.S. ambassador Nikki Haley said.
Late on Tuesday, an Islamic Jihad spokesperson said a ceasefire agreement had been reached, and on Wednesday senior Hamas official Khalil al-Hayya also spoke of an accord. Explosions shook the Palestinian enclave and smoke rose from areas hit.
There was no immediate claim of responsibility, but there was speculation the mortars were fired by Islamic Jihad in revenge for a recent incident that left three of its members dead. An Islamic Jihad spokesman said "the blood of our people is not cheap".
Several militant projectiles were shot down by Israel's Iron Dome rocket interceptor system, others landed in empty lots and farmland. The Israeli military said some of the mortar rounds were supplied by Iran. The enemy targets were rocket launchers, a lathe for manufacturing rockets, drone storage facilities, military compounds, training camps, and shops that manufacture weapons and components for anti-aircraft missiles.
Off Gaza's coast on Tuesday, the Israeli navy intercepted a boat that organisers of the Palestinian border protests launched from the enclave in a challenge to an Israeli maritime blockade.
Palestinians and their supporters say most of the protesters were unarmed civilians and Israel was using excessive force against them.
Israel withdrew its troops and settlers from Gaza in 2005 but, citing security concerns, maintains tight control of its land and sea borders, reducing its economy to a state of collapse.
They peaked on May 14, when at least 61 Palestinians were killed as tens of thousands of Gazans protested on the same day of the USA transfer of its embassy in Israel from Tel Aviv to Jerusalem.
Low level demonstrations and clashes have continued ever since.
Dozens of Palestinians have been killed by Israeli fire in the unrest.
The high Palestinian death toll in the border protests has drawn strong worldwide criticism of Israel, with rights groups saying Israel's use of live fire is illegal because in many cases it has struck unarmed protesters who did not pose an imminent threat to Israeli soldiers.
It was the first time the armed wing of Hamas has claimed responsibility for rocket attacks out of Gaza since the 2014 war.
No Israelis have been killed during that time.