The move, certain to anger Israel and the US, paves the way for the court to take jurisdiction over crimes committed in Palestinian lands and investigate the conduct of Israeli and Palestinian leaders over more than a decade of bloody conflict.
"They attack us and our land every day, to whom are we to complain? The Security Council let us down - where are we to go?" Abbas told a gathering of Palestinian leaders in remarks broadcast on official television.
"We want to refer to international institutions, and this is one we are referring to, and we'll complain to these people," he added, before signing the documents.
In the months leading up to the failed UN bid on Tuesday, Sweden recognized Palestinian statehood and the parliaments of France, Britain and Ireland passed non-binding motions urging their governments to do the same.
Israeli Prime Minister Benjamin Netanyahu said the move would expose the Palestinians to prosecution over support for what he called the terrorist Hamas Islamist group.
"We will take steps in response and defend Israel's soldiers," Netanyahu said in a statement.
Palestinian officials on Tuesday said American opposition made inevitable the defeat last night of a UN Security Council resolution calling for the establishment of a Palestinian state by late 2017 after no more than a year of peace negotiations.
The United States and Australia voted against the bid, while eight countries voted yes and another five abstained. The Palestinians were unable to achieve a hoped-for nine votes which would have forced the US to exercise its veto as one of the council's five permanent members, in a potentially embarrassing move.
Other agreements approved by Abbas included several articles on the court's jurisdiction, commitments against banned weapons and cluster munitions along with less controversial pledges on the political rights of women, navigation and the environment.
Palestinians seek a state in Gaza, the West Bank and East Jerusalem - lands Israel captures in the 1967 Middle East War.
Peace talks mediated by the United States collapsed in April in a dispute over Israeli settlement-building and a prisoner release deal, as well as Abbas's decision to sign on to over a dozen previous international texts Israel saw as a unilateral move the contravened the negotiations.
Momentum to recognize a Palestine has built up after Abbas succeeded in a bid for de facto recognition of Palestinian statehood at the UN General Assembly in 2012, which made Palestinians eligible to join the ICC.