In his first speech to the world body - one most Israelis view as a bastion of anti-Israeli sentiment - Sharon also said Israel bore no more responsibility for Gaza following the withdrawal it completed on Monday after 38 years of occupation.
"The Palestinians will always be our neighbours. We respect them and don't aspire to rule over them. They also deserve freedom and a sovereign national entity in their own country," Sharon said on Thursday, speaking in Hebrew.
But he added: "Now it is the Palestinians' turn to prove their desire for peace."
The withdrawal of Israeli troops and 8500 settlers from Gaza is part of Sharon's plan to disengage from conflict with the Palestinians.
Window of opportunity
Sharon said the pullout opened a "window of opportunity" for moving along a US-backed peace "road map" that envisages creation of a Palestinian state alongside a secure Israel.
But he said the Palestinian Authority faced its "greatest test" and must first, under the road map, "put an end to terror and its infrastructure, eliminate the anarchy of armed gangs and cease incitement" against Israel and the Jews.
"The Palestinians will always be our neighbours. We respect them and don't aspire to rule over them"
Israeli Prime Minister
Palestinian leaders said in response the only solution was a complete Israeli withdrawal from all occupied territories including the West Bank and Arab East Jerusalem, captured by Israel in the 1967 Middle East war.
"The problem can only be solved by ending the occupation that began in 1967," Palestinian chief negotiator Saeb Erekat said.
Erekat also called anew on Israel to resume negotiations on Palestinian statehood, a step Israel has ruled out until Palestinians disarm fighters opposed to peacemaking.
Confront or co-opt?
Palestinian President Mahmoud Abbas, who declared a ceasefire along with Sharon in February, has avoided confronting powerful resistance groups, citing the risk of civil war. Instead, he wants to co-opt them into security services.
The groups have refused to disarm and threatened to discard the truce in the face of Israel's continued expansion of much larger settlements in the West Bank, gobbling up occupied land key to Palestinians hopes for a state of viable size.
In his UN speech, Sharon vowed to complete construction of a West Bank barrier that Israel calls a bulwark against bombers and Palestinians condemn as a land grab.
"This fence saves lives," Sharon said about the razor wire-tipped fences and towering concrete walls that snake through occupied land.
Sharon's decision to address the UN in Hebrew and deliver the speech during prime time back home was widely seen as part of an effort to fight off a challenge by hardliner Benjamin Netanyahu for leadership of the Likud party.
Sharon defended the separation
barrier, saying it saves lives
The prime minister had particularly tough words for the UN and Iran and its nuclear programme, although he did not mention the Islamic republic by name.
Accusing the UN of passing "dozens of unjust resolutions" against Israel over the years, Sharon said: "Even today, delegates of a country, whose leaders call for the destruction of Israel, sit in this body and no one protests.
"Attempts by this state to arm itself with nuclear weapons should cause anyone who seeks peace and stability in the Middle East and throughout the world to lose sleep," he said.
Iran says its nuclear programme is purely for peaceful purposes.