Abbas, who was in New York to attend the United Nations General Assembly, held a series of talks with elements in the Security Council, asking them to bring about a halt in Israel's activities in the Strip.
Abbas' condemnation came on the backdrop of harsh criticism from Hamas directed at the Palestinian government in Ramallah.
Criticizing the Palestinian Authority in Ramallah for not responding to recent incidents in the Strip, Hamas spokesman Sami Abu Zuhri said earlier, "It is as good as cooperating with the occupation. The PA's silence proves that it is cooperating in order to eliminate Hamas in the Strip."
The spokesman continued to urge the international community to put an end to the Israeli aggression and said Israel "does not give much importance to the November peace conference".
"Israel will pay a hefty price for its aggression in Gaza," Abu Zuhri said, adding that Palestinian organizations would do everything in their power to resist Israeli attacks in the Strip.
Series of incidents
The most recent IDF airstrike took place around 2 am Thursday, when IAF jets attacked gunmen who were spotted near Beit Hanoun at a site from which Qassam rockets have been launched into Israel.
Palestinians reported that two Hamas gunmen were killed in the attack and that five others were injured.
On Wednesday afternoon, sources reported that five members of a small terror group linked to the kidnapping of an Israeli soldier last year were killed in an Israeli airstrike in the Strip.
Missiles fired by an Israeli aircraft slammed into a jeep as it traveled in the Zeitun neighborhood carrying members of the Army of Islam, the terror group implicated in the kidnapping of Corporal Gilad Shalit and BBC journalist Alan Johnston.
Palestinian sources said one of the gunmen killed in the blast was Khatab Al-Makdisi, the group's spokesperson, whom Israel accuses of links to al-Qaeda.
In a seperate incident, four Palestinians, including a child, were killed and 20 others were injured when a tank shell hit a house in the Beit Hanoun neighborhood in northern Gaza.
Eyewitnesses said 20 IDF tanks and bulldozers entered the northern Gaza Strip on Wednesday afternoon. The incursion appears to be in response to rocket from the area earlier Wednesday.