In early January, Israel froze the monthly transfer of funds it collects on behalf of the PA as a punitive measure after the Palestinians moved to join the ICC.
But on Friday, Israel agreed to release the funds after deducting debts due for electricity, water and medical services.
Speaking Sunday, Abbas confirmed that two thirds of the money had been transferred.
"They said they were going to send the money and in the end they did, but a third of it was deducted -- why?" he asked during a speech in the West Bank city of Ramallah.
"Now we have a new file to take to the ICC, first there was the (summer) war in Gaza, then there was the settlements and now the Palestinian leadership is considering presenting this issue to the court in due time."
The Palestinians would not accept anything but the full amount, he said.
"We will not take the money until we get all of it: either you give us the full amount or we go to the ICC."
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When the announcement was made on Friday, Israel did not say how much money would be transferred nor whether it would resume the normal monthly payment of around $127 million in customs duties.
The move comes after heavy international pressure on Israel to release the monthly funds, which account for two-thirds of the Palestinians' annual budget, excluding foreign aid.
A source close to Israeli Prime Minister Benjamin Netanyahu told AFP Sunday that "a portion of the money" had indeed been transferred to the Palestinian Authority but it was sent back.
The source added that Israel had "informed the United States that it is ready to transfer to the Palestinian Authority the amount that was returned whenever the PA desires it."
Under an economic agreement between the sides signed in 1994, Israel transfers to the PA tens of millions of dollars each month in customs duties levied on goods destined for Palestinian markets that transit through Israeli ports.
Although the sanction has been imposed many times, it has rarely lasted more than one or two months, except in 2006 when Hamas won a landslide victory in Palestinian legislative elections and Israel froze the funds for six months.
Blocking the money also prevents the PA paying its roughly 180,000 employees, which costs almost $200 million (170 million euros) a month.