"With regard to dealing with the Palestinian organisations, this is our affair," Abbas said in the town of Rafah on Gaza's border with Egypt on Wednesday.
"We know more and are more capable than others in dealing with our brothers."
Ministers of the Quartet - United States, the United Nations, Russia and the European Union - said in a joint statement on Tuesday that following Israel's pullout from Gaza, Palestinians needed to "dismantle terrorist capabilities and infrastructures".
Israel and Washington have long demanded Abbas disarm resistance fighters in order to help restart peace talks.
The moderate Palestinian leader has preferred to co-opt resistance fighters with jobs after having coaxed them into a ceasefire in February.
Abbas believes that to try to crush the resistance would risk civil war while Palestinians remain far short of achieving a state on lands occupied by Israel in a 1967 war.
While jettisoning the small Gaza Strip, Israel continues to expand much larger settlements in the West Bank that it vows never to cede under any peace treaty.
Hamas has portrayed Israel's
pullout as a victory for their fight
They straddle land at the heart of Palestinian aspirations to a state of viable size.
Resistance fighters rose to dominance in the streets during an uprising against Israel and have flexed their power in the wake of the Gaza pullout with marches by thousands of heavily armed men, who portrayed the Israeli departure as a victory for their fight.
They have said the ceasefire could fall apart if Israel keeps tightening its grip on the West Bank.
The Islamist movement Hamas, a key resistance group, plans to run candidates in a Palestinian parliamentary vote next year for the first time.
Israel and Washington fear that Hamas, forecast to win up to a third of votes in the election, could use its rise towards the Palestinian political mainstream to block future efforts to reach a permanent peace agreement.
Prime Minister Ariel Sharon has said Israel could obstruct preparations for the vote if Hamas is allowed to run without first disarming and renouncing its vow to destroy Israel.
The Quartet, which in 2002 drafted a "road map" peace plan, said armed groups should not run in the Palestinian ballot as there was a "a fundamental contradiction" between what it called "militant violence" and "the building of a democratic state".
Condoleezza Rice has cautioned
Israel about Palestinian polls
But US Secretary of State Condoleezza Rice has cautioned that Israel should give the Palestinians leeway for "the evolution of their political process".
Abbas said he welcomed the Quartet's endorsement of a future Palestinian state, which is mentioned in the road map and which the Palestinians want to establish in the West Bank and the Gaza Strip with Arab East Jerusalem as its capital.
The road map demands the dismantling of resistance groups as well as a halt to Israel's enlargement of West Bank settlements.
Hamas too on Wednesday rejected the Quartet's appeal to disarm.
"We absolutely refuse the Quartet's request. There is no link between carrying weapons and participating in elections," the resistance group said in a statement.
"We reject this call, which is an interference in internal Palestinian affairs aiming to provoke inter-Palestinian fighting," it added.
"We think that the Quartet would do better to ask the enemy prime minister (Israeli premier Ariel Sharon) to put an end to the occupation of all Palestinian territories and release all Palestinian prisoners he is holding."