In a televised speech in the West Bank city of Ramallah broadcast live on Palestinian television, Abbas said he has a constitutional right to sack the Hamas-led government.
Abbas said the constitution gives him the right to fire the government "whenever" he wants.
The Hamas-led government immediately rejected Abbas's call. The government said in a statement that it considers it "a coup against Palestinian legitimacy and the will of the Palestinian people."
Foreign Minister Mahmud Azzahar said Hamas will do nothing to cooperate with possible early elections, saying Hamas will not participate or allow elections to be held.
"The government will not arrange for any election and the central committee for the election should cooperate," Azzahar said. "If we are not going to cooperate, there will be, practically, no election."
The call was followed by demonstrations attended by thousands of Hamas supporters. Hamas gunmen exchanged gunfire in Gaza with Fatah forces loyal to Abbas. At least five people were wounded.
The United States and Britain welcomed Abbas's call for early elections. A White House spokeswoman said the U.S. government hopes fresh elections will help get the peace process back on track. British Prime Minister Tony Blair also supported Abbas's call.
Israeli government spokeswoman Miri Eisin said Israel has "clearly stated" its support for Palestinian moderates "such as" Abbas.
Russia, meanwhile, has urged the two sides to seek a compromise.
The speech comes after escalating violence between Hamas supporters and Fatah forces loyal to Abbas in the West Bank and Gaza.
Abbas said he blames Hamas for the current Palestinian crisis, and that a political solution is needed to ensure security and stability.
He said he wants an administration that can lift Western sanctions. The sanctions were imposed on the Hamas-led government after it refused to recognize Israel.
Abbas also denied any conspiracy to kill Prime Minister Ismail Haniyah after his convoy was attacked earlier this week.
(AFP, AP, Reuters)