Abbas, however, has ruled out talks with Hamas which he has accused of trying to assassinate him and of carrying out a coup in Gaza.
Hamas called Abbas's remarks "disgusting" and thousands of Hamas supporters denounced him in the streets of Gaza on Thursday.
In a television address a day earlier, Abbas criticised Hamas for attacking "national symbols" during its takeover of Gaza last week.
He accused Khaled Meshaal, Hamas leader-in-exile, of being part of a plot to assassinate him, referring to the group's members as "murderous terrorists".
Abbas also told Fatah leaders in the occupied West Bank that there would be "no dialogue with those killers, coup plotters".
'Project of darkness'
Abbas said Hamas replaced the "national project" with "its project of darkness", attacking the symbols of government in Gaza, including the house of the late leader Yasser Arafat.
It was Abbas's toughest speech since he fired the Hamas-led cabinet and replaced it with his own team of Fatah supporters and experts over the weekend.
"Our main goal is to prevent sedition from spreading to the West Bank, ... to prevent violations by any party, and to deal [with everyone] equally, based on law," Abbas said.
He accused Hamas of trying to set up its own state in Gaza alone, a step he said would scuttle Palestinian hopes for independence.
He said he had tried to prevent the conflict through "continuous dialogue". Instead, "we are seeing assassination of leaders of Palestinian security and Fatah in Gaza".
"It's a fight between the national project and this small kingdom they want to establish in Gaza, the kingdom of Gaza, between those who are using assassination and killing to achieve their goals, and those who are using the rules of law."
'Assassination plot'
Abbas accused Hamas of trying to assassinate him when he planned a visit to Gaza a month ago, digging a tunnel under a road where his car was to pass and trying to fill it with 250 kilograms of explosives.
He said he received videotapes of the operation, showing armed groups with Hamas signs on their shirts carrying out the work.
He dismissed Hamas claims that the explosives tunnel was aimed at Israelis.
"I have sent these tapes to all the Arab countries, to show how much this dark movement is acting," he said.
"The coup seekers through their madness have given a golden opportunity to those who want to separate Gaza from the West Bank," he said.
Abbas appealed to the Palestine National Council, a PLO body, to give his new government its support.
Hamas reaction
Sami Abu Zuhri, a Hamas spokesman, rejected Abbas's statements.
"What he said was disgusting and not appropriate for the Palestinian president," the Hamas official said.
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Thousands of Hamas supporters gathered in
Gaza chanting anti-Abbas slogans [AFP] |
"The president has harmed himself with his words."
Ayman Taha, another Hamas official, told AFP that the speech was "a joke, contains a lot of contradictory statements, is misleading and a pack of lies".
Not long after Abbas's speech, Hamas backers on the streets of Gaza city denounced him as an agent of Israel and the US.
Thousands of people chanted anti-Abbas slogans and rallied in support of the Hamas movement. They also burned US and Israeli flags, and pictures of Abbas and Condoleezza Rice, the US secretary of state.
Also on Wednesday, Tzipi Livni, the Israeli foreign minister, spoke to Salam Fayyad, the new Palestinian prime minister, to begin for a "a dialogue between the two governments," according to Livni's spokesman.
The Israeli prime minister, who is visiting Washington for talks on a range of issues including extending US military funding for another 10 years, has said he wants "every possible effort" to co-operate with Abbas but has refused to agree to renewed peace talks yet.