In an interview with Newsweek magazine to be published on Monday, Abbas said the Palestinian Authority would ensure that Hamas did not take over the settlements, which the Israeli army left on Sunday after a 38-year presence in Gaza.
"We are ready to control the whole security situation," Abbas told the magazine. "We will not allow anybody to rush into the settlements."
He said the next step in the peace process with Israel was to return to the stalled, internationally backed road map peace plan.
"There was an agreement that the Israelis would evacuate [some West Bank] cities ... . This was not implemented. We talked about prisoners, and they didn't release any.
"Of course, we want them to freeze [West Bank] settlement activities and [the building of] the wall."
In a separate interview, Israeli Prime Minister Ariel Sharon told the magazine that he believed his controversial plan to withdraw Jewish settlers from Gaza had strengthened Israel's hand.
"I think the position of Israel in the world is much better right now ... I would like to make a major effort to solve the problems between us and the Arabs," Newsweek quoted him as saying.
But in order to return to the road map, he said, "there should be a full cessation of terror, hostilities, and incitement".
"We are ready to control the whole security situation. We will not allow anybody to rush into the settlements"
"The Palestinian Authority should dismantle the terrorist organisations ... . When we leave [Gaza], our reaction if terror continues will be very, very hard," he warned, insisting that Abbas shouldered that responsibility.
"If he will do his part, I think we can do many things, but he must do his part," Sharon said.
At a press conference held on Sunday afternoon in Gaza City, Palestinian Minister of Civil Affairs Muhammad Dahalan accused Israel of imposing its will on the Palestinians regarding the Rafah crossing.
"Everyone knows Israel has a clear plan to force us to agree on the Israel proposal to close the Rafah crossing and move it to Kerem Shalom.
Palestinians fear Israeli border
controls will make all Gaza a jail
"There has been no agreement on the issue of border control yet and we blame Israel fully for leaving Gaza imprisoned," he said.
"Our technical teams put all the options on the table for a comprehensive vision of the disengagement, including the border crossings, but they were rejected by Israel.
Dahalan added: "Israel holds full responsibility for the current border crisis."
Dahalan said the Palestinian position had not changed - that the current crossing must remain intact and under Palestinian control.
"If the Israelis think they will leave and close this place up they are mistaken. If they think the Palestinians will remain quiet in the face of collective imprisonment, they are mistaken"
Dahalan: A new crossing will add
to Palestinian suffering
"Our position is clear: We absolutely refuse that the current crossing be moved to anywhere else. If it is moved to Kerem Shalom, this will mean greater suffering for Palestinians.
"The crossing has to be fully Palestinian-Egyptian. We refuse any other proposal."
Dahalan also warned Israel against evacuating Gaza and leaving it as a large prison.
Responding to a question about the fate of the evacuated synagogues, Dahalan said Israel no longer considers them synagogues but buildings, without elaborating on what would be done with them.
Diana Butto, legal adviser to the Palestinian committee on evacuation, said the only viable solution was one in which Palestinians maintain complete control of the current crossing.
Butto said the unilateral Israeli decision is designed to put pressure on the Palestinian Authority to accept the Israeli proposal and will have serious repercussions on daily life in Gaza.
"They are trying to pressure the Palestinians - they are trying to put the burden on us to solve this border crisis," Butto told Aljazeera.net.
"The closure [of Rafah crossing] means that Palestinians seeking to travel out of the Gaza Strip or come in are effectively denied their freedom of movement"
legal adviser to Palestinian Authority
"The implications are obviously very serious - the closure means that Palestinians seeking to travel out of the Gaza Strip or come into the Gaza Strip are effectively denied their freedom of movement.
"This of course will impact not only on individual travel but in the cases of humanitarian conditions, travel will be onerous and tedious ... [and] will not improve."
Aljazeera.net's Laila El-Haddad contributed to this report from Gaza