Membership of the Rome Statute, which governs the ICC, would open a way for Palestinians to pursue criminal complaints in The Hague, but is firmly opposed by both Israel and the United States.
"This is a very significant step," envoy Riyad Mansour told reporters. "We are seeking justice for all the victims that have been killed by Israel, the occupying power."
UN Secretary General Ban Ki-moon is to review the letters and notify state members on the request within 60 days.
The United States has said it strongly opposes the move, branding it "counterproductive" to efforts to restart the peace process with Israel.
But the Palestinians hope joining the court will pave the way for them to seek justice against Israel for its actions in the occupied territories.
Mansour said the Palestinians had also filed a request with the ICC in The Hague to grant retroactive jurisdiction to the court to cover alleged war crimes committed during last year's Gaza war.
The Palestinians were upgraded from observer entity to a UN "observer state" in 2012, opening the possibility for them to join the ICC and a host of other international organisations.
Israeli Prime Minister Benjamin Netanyahu urged the Hague-based ICC to reject the request, arguing that the Palestinian Authority (PA) did not rank as a state.
But ICC chief prosecutor Fatou Bensouda has already said that Palestine qualifies to join the Rome Statute.
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