In a speech at a Palestinian hospital in the Egyptian capital of Cairo, Abbas lashed out at the Islamist movement for its conduct during the three-week Gaza war, accusing the group of "irresponsible behavior" and of carrying out "an agenda which is not Palestinian."
"Now we say ... no dialogue with those who reject the Palestine Liberation Organization," Abbas told a news conference. "They must admit without equivocation or ambiguity that the organization is the sole and only representative of the Palestinian people.
"Then there will be dialogue," he added.
In Syria, exiled Hamas officials were quick to answer back.
"We want to tell Abbas that we are not begging for dialogue and we are not running after it," senior Hamas leader Mohammed Nazal told CBS News in Damascus.
The Islamic faction has ruled Gaza since seizing power there after elections in June 2007.
"Anyway, we were not surprised to hear Abbas's inflammatory tone and void statements which contained false accusations, as that reflects his state of bewilderment and confusion after the great victory of Hamas and other resistance groups against Israel," Nazal said.
"Abbas and his team were waiting to return to Gaza on the Israeli tanks after most of them ran away in their underwear to Ramallah and Cairo," he added.
The remarks come as Palestinians from both sides were gathering in Cairo for talks on Monday aimed at bolstering a truce in Gaza.
Egypt has been mediating a truce after Hamas and Israel announced ceasefires on January 18, ending the devastating 22-day offensive that killed more than 1,330 Palestinians — half of them women and children — and wounded 5,000 others.
Thirteen Israelis were killed during the same period.
Hamas leader Khaled Mishaal (visiting Tehran to thank the Iranian government for its support during the offensive) has advocated a new umbrella body to represent the diaspora of Palestinians in Gaza, the West Bank and beyond, calling for "a new, national authority."
The two sides agree in principle on the idea of a national unity government for the Palestinian Authority, but they disagree on whether Abbas still has a mandate to govern, and on whether armed struggle is still a legitimate strategy for dealing with Israel.
By CBS News' George Baghdadi reporting from Damascus.