Using tanks, warplanes, and artillery fire, on Friday the Israeli army launched a major attack on the Gaza Strip, allegedly in response to gunfire towards its soldiers along the border, according to Israeli sources.
A third man was killed by Israeli fire east of Rafah, also in southern Gaza, the ministry said.
The Israeli military says the attacks were in response to shots fired at its troops from Gaza.
The protests are largely prompted by the Israeli-Egyptian blockade, which was imposed after the militant group assumed control of Gaza in 2007.
Hamas, the enclave's Islamist rulers, has reported three members of its military wing killed; Shaban abu Khatar, Mohammed abu Farhana and Mahmoud Qushta, saying they were fighters. The Presidency said in a statement that Abbas started calls with regional and global sides to prevent escalation of the situation.
The same day, around 200 rockets and mortars were fired at Israel from Gaza and four Israelis were wounded when a rocket hit a house in the nearby Israeli city of Sderot.
Palestinian residents said the area was calm after midnight, and an Israeli Army spokeswoman confirmed there was no known military activity in the Gaza Strip.
Tension along the fence has been simmering for months, at times erupting into violence.
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At least four Palestinians were killed in later Israeli strikes, according to Palestinian medical officials.
Palestinians are protesting to demand the right to return to homes and villages they fled or were driven from during the conflict surrounding the creation of the modern state of Israel in 1948.
Israel has said it is defending its sovereign border.
"Earlier today, IDF aircraft and tanks targeted military targets belonging to the Hamas terror organisation".
"Those who want to provoke Palestinians and Israelis into another war must not succeed", he wrote on Twitter.
Israel resumed air attacks on Gaza after last week's projectile attacks, warning that it would take whatever action necessary to stop incendiary kites and balloons being flown over the border.
Khalil al-Hayya, a top Hamas leader, said "the protests will remain, flaming and existing, and its tools will multiply and diversify, including kites, until our goals are achieved".