Hamas quickly hit back, describing the allegations as "lies".
In a speech marking the 10th anniversary of the death of veteran Palestinian leader Yasser Arafat, Abbas said Hamas was behind the Gaza explosions which targeted leaders of his Fatah movement.
The blasts prompted the cancellation of a rare memorial service in Gaza for Arafat, the Fatah founder who died in mysterious circumstances in a hospital near Paris.
"Those who caused the explosions in Gaza are the leaders of Hamas -- they are responsible," Abbas said in the West Bank city of Ramallah, accusing the rival faction of trying "to sabotage and destroy the Palestinian national project."
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The deal led to the creation of a government of national consensus which took office in Ramallah but has yet to fully exert its powers in Gaza, Hamas's stronghold.
Following his speech, Hamas denounced Abbas as "sectarian and partisan."
"Abbas's speech is web of lies, insults and disinformation," said Mushir al-Masri, a Hamas spokesman in Gaza.
"What the Palestinian people need is a courageous president."
Abbas also addressed the clashes which have gripped annexed east Jerusalem for the past four months and spoke about unrest at the flashpoint Al-Aqsa mosque compound triggered by the demands of far-right Jewish fringe groups for the right to pray there.
Abbas also reaffirmed his plans to submit a draft resolution to the UN Security Council later this month calling for an end to Israel's occupation of the Palestinian territories by November 2016.
He promised that the Palestinians, who won the UN rank of observer state in 2012, would apply to join a host of international organisations if the resolution was blocked by a US veto.
"We will not tolerate any pressure," he said, referring to US efforts to dissuade the Palestinians from approaching the Security Council.