Abbas thanked the Arab and Islamist states, which he said had always stood by the Palestinians and the Palestinian issue.
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The Palestinian president promised to continue the national struggle, "until we wave the Palestinian flag over east Jerusalem."
According to Abbas, the UN resolution was a victory for peace, freedom and international law.
Shortly after Prime Minister Benjamin Netanyahu praised the eight countries which voted with Israel against the Palestinian bid, Abbas thanked the 138 countries that voted in favor of upgrading the status of Palestine. Forty-one countries abstained in the General Assembly vote.
Palestinian Prime Minister Salam Fayyad expressed his satisfaction with the diplomatic move as well, saying that "this achievement has greatly benefitted the efforts invested by the PA on all levels in order to prepare for the establishment of a state."
He noted that the international community must implement what the international law requires in order for the Palestinian people to obtain their rights, freedom and independence.
"The PA is in the final stage before becoming an independent state capable of providing services to its people."
Hamas counting on 'heroic resistance'
Hamas politburo chief Khaled Mashaal told Reuters that the de facto recognition of a sovereign Palestinian state should be seen alongside Gaza's latest conflict with Israel as a single, bold strategy that could empower all Palestinians.
"I told Abu Mazen (Abbas) we want this move to be part of a national Palestinian strategy" that includes "the (armed) resistance which excelled in Gaza and gave an example of the ability of the Palestinian people to resist and steadfastly confront the occupier," a confident Mashaal said.
Mashaal, who will visit the Gaza Strip next week, said the short war which claimed 162 Palestinian lives and five Israelis was concluded on terms set by the Palestinian Islamist movement and ended its isolation, creating a new mood that could lead to reconciliation with Abbas' Fatah.
He compared Israel's mood of dejection with the jubilation of Palestinians in Gaza and across the West Bank led by Abbas, insisting that "for the first time a ceasefire was achieved on conditions set by Hamas, and in the presence of the Americans."
The Hamas prime minister in Gaza, Ismail Haniyeh, praised the "diplomatic victory," linking the Israeli operation in Gaza to Thursday's vote.
"What happened at the General Assembly is the height of the resistance and sacrifice of the Palestinians, and confirms the victory in Gaza. We must persist with the resistance and jihad."
Izzat Rishaq, a senior Hamas figure in exile, said he welcomed the UN vote as an achievement, but that Hamas counted on "heroic resistance" to create a Palestinian state – underlining the group's deep ideological rift with Abbas who opposes violence.
The Islamic Jihad chose to downplay the resolution. Several hours before the vote, when the results were already known, the organization's Secretary-General Ramadan Salah said his movement did not welcome the Abbas plan but would not lash out at it.
"What Palestine are we talking about? If we were talking about the whole Palestinian homeland from the river to the sea, we would naturally welcome it, but if we are talking about setting a ceiling for the Palestinian rights (the 1967 borders), then as we said sincerely to the PA and our bothers in Egypt – we don't welcome this plan but won't act against it."
Reuters and The Associated Press contributed to this report