Mr Abbas said he was furious over the bomb that targeted the convoy carrying his prime minister, Rami Hamdallah, and security chief last week, which did not seriously injure anyone.
Hamas added that his remarks "undermine the prospects of advancing the Palestinian national project and achieving national unity".
"In light of all this, Hamas calls for general elections, including presidential, parliamentary and national council elections, so that the Palestinian people can choose their leadership".
Over the last three months, Abbas has said that Trump's decision to recognize Jerusalem as Israel's capital and move the USA embassy there from Tel Aviv as the "slap of the century" - a move that, in Abbas' view, disqualifies him from any role in future peace talks between the PA and Israel.
The Jerusalem Post reported that Friedman managed to read about Mahmoud Abbas's canine curse just minutes before the diplomat addressed the Sixth Global Forum for Combating Antisemitism.
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Hamas condemned what it called Abbas's "provocative positions", saying its security services were still investigating the explosion.
Fatah and Hamas have tried for years to come to an accommodation over running the Gaza Strip, but have repeatedly failed to implement deals mainly brokered by Egypt.
"We do not want an investigation from them, we do not want information from them and we do not want anything from them because we know exactly what they, the Hamas movement, were the ones who committed this incident", he said.
He said he would take new punitive measures against Hamas.
"Imposing the sanctions will lead to the collapse of the Gaza Strip", the PFLP warned.
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Azevedo is in India for a two-day informal ministerial meeting of WTO members over Monday and Tuesday. This is the second WTO mini-ministerial meeting being hosted by India, after the first in 2009.
Abbas spoke in Ramallah, expressing anger at US Ambassador to Israel David Friedman's criticism of the Palestinian Authority after it refused to condemn the recent terror attacks in which Jews were murdered. The embassy added that it is working with the Israeli authorities.
"Antisemitism or political discourse? I'll leave that all up to you".
In January, Abbas railed against the White House, cursing Trump, saying, "May God demolish your house". "For the first time in decades, the U.S. administration has stopped spoiling the Palestinian leaders and tells them 'enough is enough.' Apparently the shock of the truth has caused them to lose it".
Speaking during Monday's meeting, Abbas said that if the "assassination attempt" had succeeded, the development would have opened the door for a bloody civil war. "Apparently the shock of the truth has caused them to lose their mind", he said.
Rabada appeal hearing set for Monday
I'm sure he'll expect them to say something on the field and he'll be prepared for that". Australia's tour has been marred by altercations on and off the field.