Palestinian President Mahmoud Abbas said Israel's "fierce aggression" on Gaza "indicates that the occupation does not want peace".
In a follow up announcement, Israel's Minister of Energy and Water Resources, Yuval Steinitz, instructed the IEC not to put its employees in danger to fix the damage.
"They have the ability, the control and the power to escalate or to deescalate the situation, to rein in the Palestinian's and their own extremist factions in Hamas or to escalate the situation", said military spokesman Jonathan Conricus.
"Qassam and Jerusalem Brigades (the groups' armed wings) announce joint responsibility for bombarding (Israel's) military installations and settlements near Gaza with dozens of rocket shells throughout the day", they said in a joint statement.
The Israeli military struck dozens of militant sites in Gaza overnight as rocket fire continued toward southern Israeli communities into early Wednesday morning, setting off air raid sirens in the area throughout the night.
An Islamic Jihad spokesman said "the blood of our people is not cheap".
The protests peaked May 14, the same day the USA opened its embassy in Israel's capital, Jerusalem, when more than 60 people died in a Hamas-organized action to breach the Israeli border and return to homes that were lost 70 years ago during the war that established Israel.
Approximately 25 of the mortars were intercepted by Israel's missile defence system, he said.
Israel's military said it was not seeking an escalation, but warned Hamas.
The high Palestinian death toll in the border protests has drawn strong global 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.
U.S. to Impose Limits on Some Chinese Visas
The White House also says the USA is planning new investment restrictions and export controls. China has pledged to retaliate against any tariffs imposed by the US.
Danny Danon, Israel's permanent representative to the United Nations, blasted the proposed resolution calling it "shameful" and crafted to help Gaza's rulers Hamas.
Three Israeli soldiers were wounded in the exchange of fire.
Prime Minister Benjamin Netanyahu pledged a forceful response to the barrage of mortar shells fired.
The Israel-Gaza border fell quiet on Wednesday under an Egyptian-brokered ceasefire after the most intense flareup of hostilities between Palestinian militants and Israel since a 2014 war.
Palestinians and their supporters said most of the protesters were unarmed civilians and Israel was using excessive force against them.
Israel has explained its actions citing security reasons, saying it is defending its borders and blames Hamas for encouraging thousands of Palestinians to break through the border and attack Israelis.
Kuwait, a non-permanent council member that represents Arab countries, circulated a draft UN Security Council resolution calling "for the consideration of measures to guarantee the safety and protection of the Palestinian civilian population" in the occupied Palestinian territories and the Gaza Strip.
The military said the vessel was stopped without much incident and would be towed to the Israeli port of Ashdod where the 17 people on board would be questioned and then returned to the Gaza Strip.
At least 116 Palestinians have been killed by Israeli fire and more than 10,000 have been injured since mass demonstrations began on March 30. These attacks came after months of using Palestinian human shields as cover for other attacks on Israel.