The move comes after the Security Council adopted a resolution demanding an end to Israeli settlement building on occupied land.
The resolution was passed after the US refused to veto it, breaking with long-standing American practice.
Washington has traditionally sheltered Israel from condemnatory resolutions.
Blame and bitterness keeping peace at bay
Mr Netanyahu insists that Israel will not abide by Friday's vote at the 15-member UN Security Council, which was welcomed by Palestinian leaders.
"I instructed the Foreign Ministry to complete within a month a re-evaluation of all our contacts with the United Nations, including the Israeli funding of UN institutions and the presence of UN representatives in Israel," he said.
He described the Security Council decision as "biased and shameful", adding: "It will take time, but this decision will be annulled."
The Egyptian-drafted resolution had been withdrawn after Israel asked US President-elect Donald Trump to intervene, but it was proposed again by Malaysia, New Zealand, Senegal and Venezuela.
The resolution, approved by 14 votes to zero, with only the US abstaining, demands that Israel immediately "cease all settlement activities in the occupied Palestinian territory, including East Jerusalem".
It says Jewish settlements are a "flagrant violation under international law and a major obstacle to the achievement of the two-state solution and a just, lasting and comprehensive peace".
Israel earlier announced that its ambassadors to New Zealand and Senegal had been ordered to return for consultations and that it was cutting all aid programmes to Senegal.
Israel has no diplomatic ties with Malaysia and Venezuela.
The issue is one of the most contentious between Israel and the Palestinians.
About 500,000 Jews live in about 140 settlements built since Israel's 1967 occupation of the West Bank and East Jerusalem. The settlements are considered illegal under international law, though Israel disputes this.
A spokesman for Mr Abbas said: "The Security Council resolution is a big blow to Israeli policy, a unanimous international condemnation of settlements and a strong support for the two-state solution."
The Palestinian ambassador to the UN, Riyad Mansour said: "The Council's action, while long overdue, is timely, necessary and important."
The US envoy to the UN, Samantha Power, said the resolution reflected the "facts on the ground" that settlement growth had been accelerating.
"The settlement problem has gotten so much worse that it is threatening the two-state solution," she said.
Criticising Mr Netanyahu, she said: "One cannot simultaneously champion expanding settlements and champion a two-state solution that would end the conflict."
However, she said the US had not voted in favour of the resolution because it was "too narrowly focused" on settlements.
Meanwhile, Mr Trump, who will be inaugurated on 20 January, tweeted after the vote: "As to the UN, things will be different after Jan. 20th."
On Thursday, Mr Trump had urged the council to reject the motion.

The European Commission says it is Europe's most comprehensive study so far of bee colony deaths.
Winter mortality was especially high for bees in Belgium (33.6%) and the UK (29%) in 2012-13. But in spring-summer 2013 France was highest with 13.6%.
Bumblebees and other wild bees were not studied, nor were pesticide impacts.
The study, called Epilobee, described 10% as an acceptable threshold for bee colony mortality - and Greece, Italy and Spain were among the countries with rates below that threshold.
The mortality percentages are national estimates based on representative samples. All 17 countries applied the same data collection standards, the report says.
The survey covered almost 32,000 bee colonies.
But there is also much concern about death rates among wild bees, which are vital pollinators too.
Last year the EU introduced a ban on four chemicals called neonicotinoids which are used in pesticides.
They are believed to be linked to the collapse of bee colonies across Europe, though there is a heated scientific debate over the chemicals' impact and many experts say further studies are needed.
The Commission wanted pesticide impacts to be included in the Epilobee study, but it was overruled by member states' governments.
Israeli Prime Minister Benjamin Netanyahu has said Israel will re-assess its ties with the United Nations.
[NEXT_CONCEPT]
A new study covering 17 EU countries says that far more honeybees are dying in the UK and other parts of northern Europe than in Mediterranean countries.