The earlier firing of some 28 mortar shells toward Israel caused no injuries, with most intercepted by air defence systems, but Prime Minister Benjamin Netanyahu vowed a forceful response. No Israelis have been killed and only one soldier has been reported lightly wounded.
Nikolay Mladenov said the global community should "unequivocally condemn" Hamas' massive attack against Israel using 216 rockets and mortars.
But after Tuesday's flareup, followed by sirens and explosions into the night, there were Palestinian claims of a ceasefire.
No airstrikes by the Israeli military have been confirmed either, although Israeli officials did not say whether Israel had agreed to a ceasefire.
Now, Professor Abusada said by phone, the groups believe that their rocket and mortar fire had succeeded in sending "a crystal-clear message to Israel that if Israel targets Palestinians and bombs Gaza, it should expect Palestinian retaliation".
More than 110 Palestinians have been killed by Israeli army fire since the anti-blockade protests began in late March, including at least 60 on May 14 when Israeli troops fired live bullets and heavy volleys of tear gas from across the border fence. Three members of Islamic Jihad were killed in an Israeli strike on Sunday.
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There were conflicting statements about whether the boats would seek to break Israel's blockade, which now allows them to travel nine nautical miles (16 kilometers) off the coast. Palestinians and their supporters say a lot of the protesters have been unarmed civilians and Israel was utilizing extreme power towards them. One exploded in the yard of a kindergarten before it was due to open.
"IDF forces attacked 25 terrorist targets of the Hamas terrorist organization in the Gaza Strip", Israeli military said in a statement on Tuesday evening.
The Israeli military said it was "carrying out activities in the Gaza Strip", without elaborating.
Organizers of the Palestinian border protests launched a boat from Gaza on Tuesday in a challenge to Israel's maritime blockade of the enclave.
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. "No one has the right to play with the lives of two million people who have lived through hell in the last decade".
The boat carried activists, students and patients in need of medical treatment outside the Gaza Strip.
The Israeli army said in a statement that Israeli planes targeted 30 sites in Gaza. It accuses Gaza's Islamist rulers Hamas of seeking to use the protests as cover for attacks.