Sderot Mayor Alon Davidi called upon the IDF to respond "in full force" to the bullet fire.
"Hamas is exclusively responsible for the events transpiring in and out of Gaza, and is accountable for all terrorist activity emanating from Gaza targeting Israeli civilians and Israeli sovereignty", the IDF said.
Israel says the military has delivered eight trucks full of humanitarian aid and medical supplies to the Gaza Strip, where Israeli troops firing from across the border killed almost 60 Palestinians at mass border protests on Monday.
Palestinians seek East Jerusalem as the capital of their future state in the two-state solution, which would be established in the occupied West Bank and the Gaza Strip. "About 1,200 others suffered other types of injuries, including from tear gas, Palestinian sources said".
The Islamic militant group has emerged without tangible achievements from weekly border protests in which more than 100 Palestinians were killed and hundreds wounded by Israeli fire since late March.
Omar Shakir, the local director for Human Rights Watch, said that if they were unarmed, the identity of the slain protesters and their affiliation to a militant group has little to do with Israel's right under worldwide law to use lethal force at a demonstration.
Israel accuses Hamas, with whom it has fought three wars since 2008, of seeking to use the demonstrations as cover for violence.
"They were keen these marches do not slide into armed confrontations and we agree with the brothers in Egypt over that", Sinwar said in an interview on Al Jazeera television. Most of Gaza's residents are descendants of refugees.
"Behind the protests is Hamas, which governs Gaza and is regarded as a terrorist group by the U.S., Israel and the European Union". Israel disputes this and says some of those killed were firing at its troops.
Israeli officials quickly seized upon his remarks. Israeli military spokesman Lt Col Jonathan Conricus said Hamas had "set the record straight", adding: 'This was no peaceful protest'. "Take his word for it".
Deadly clashes have further escalated during the so-called "Great March of Return", which culminated on Monday when the USA embassy officially opened in Jerusalem. Paraguay is slated to do the same next week.
Donald Trump's decision to move the United States embassy to Jerusalem has added to Palestinian frustrations. Those talks have been stalled since 2014.
The status of Jerusalem is one of the thorniest obstacles to forging a peace deal between Israel and the Palestinians, who with broad worldwide backing want East Jerusalem, captured by Israel in the 1967 Middle East war, as their capital.