"If the government had anything to do with the decision we want it to resign," Muhammad Jadallah, the head of the Coalition for Jerusalem, said.
Thirty-two Palestinian groups in Europe also called on Abbas to immediately step down from office.
In a statement, the groups said "the step to delay the endorsement was not less dangerous than the atrocities committed by the Israeli occupation in Gaza".
Critics have accused Abbas of letting down his people by bowing to US pressure on the resolution, but Abbas dismissed the criticism on Sunday.
"The issue of postponing the vote didn't come from us - we are not members of this international organisation [UN Human Rights Council]," he said.
"I believe all the Arab brothers are members of the organisation and they all know very well that the postponement of the vote happened with their knowledge and approval."
The adoption of the report by the 49-member council was seen as a key step towards eventually bringing war crimes charges against Israeli leaders and Palestinian fighters at the International Criminal Court (ICC).
Though the 575-page report by Richard Goldstone blamed both the Israeli military and the Palestinian fighters for war crimes during Israel's offensive on the Gaza Strip between December and January, it was more critical of Israeli troops for "targeting and terrorising civilians".
But the UN Human Rights Council on Friday deferred endorsement of the report until March as requested by sponsors of the resolution. Critics of the move say the postponement came after US pressure.
Nour Odeh, Al Jazeera's correspondent in the West Bank, said: "Palestinian political factions, including the president's own Fatah movement, as well as human rights organisations, have been escalating their expressions of anger and condemnation about their leadership's decision to support the withdrawal," she said.
"Many questions are being posed about how this unfolding crisis could affect ongoing efforts to reach national reconciliation."
Hamas, the Palestinian group which runs Gaza and is political rival to Abbas's Fatah, has blamed Abbas for the delay and ruled out reconciliation under the present circumstances.
"Even if a national reconciliation deal is signed, how can it be implemented if this political policy persists?" Ismail Haniya, Hamas's leader in Gaza, said on Monday.
"How can we achieve unity if this political leadership persists? Without removing these figures, there can be no consensus because this position is not the subject of consensus."
There has been strong dissension within Fatah over the delay as well.
One Fatah official said: "The consent to defer the vote had cost us dear. We'll need years to fix this mistake."
Salam Fayyad, Abbas's prime minister, has demanded the report's recommendations be implemented in full, while Ali Jarbawi, the Palestinian planning minister, expressed his "surprise" over the consent to postpone the vote.
Abbas has also drawn criticism from Syria, which postponed his planned visit to Damascus in a gesture of protest.
During the three-week Gaza war, more than 1,400 Palestinians - one-third of them women and children - were killed while Israel lost 10 soldiers and three civilians.