War crime charges have been filed in the ICC against persons in the newly recognized Palestine by an Israeli group
Separately, an Israeli group, Shurat Hadin, filed war crimes complaints against Abbas, three other Palestinian officials and Hamas leader Khaled Mashaal. The group says it is preparing 30 more complaints that it hopes will deter Palestinians from seeking ICC action against Israel.
Palestinians are now subject to the jurisdiction of the ICC with their membership and as expected war crimes charges have begun to be filed. Additional charges are thought could be brought by Israel.
DO PALESTINIANS FACE POSSIBLE WAR CRIMES INVESTIGATIONS?
Yes. U.N. investigators say Hamas has committed war crimes by indiscriminately firing rockets and mortar rounds at Israeli civilians and launching attacks from civilian areas in Gaza. The militants, who seized Gaza from Abbas in 2007, have carried out such attacks for more than a decade and Israel says it launched the last Gaza war to stop rockets.
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Israel says Abbas and his aides could also face prosecution, arguing he is legally responsible for Hamas' actions because the group supports his Palestinian unity government.
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In joining the ICC, the Palestinians are transforming their relations with Israel from tense to hostile, reducing chances of negotiations on statehood.
The Palestinian Authority, Abbas' self-rule government, could collapse if Israel continues to freeze tax transfers and Congress cuts off funding, as threatened. This could cost Abbas his job, damage ties with the U.S. and move Palestinian independence further out of reach.
The charges and expected counter-charges from Palestine will not bring peace to the region one day closer but could easily cause further delay in reaching the much needed treaty for peace between two states.
UPDATE: In terms of standing and jurisdiction of the court the below is excerpted from wiki
The court's founding treaty, the Rome Statute, provides that individuals or organizations may submit to the Prosecutor information on crimes within the jurisdiction of the Court. These submissions are referred to as "communications". Before an investigation is opened, each communication is subject to three levels of analysis: initial review, basic reporting and intensive analysis.
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Some of the communications received by the Prosecutor alleged that crimes had been committed on the territory of states parties to the Court, or by nationals of states parties: in such cases, the Court may automatically exercise jurisdiction. Other communications concerned conduct outside the jurisdiction of states parties: in these cases, the Court can only act if it has received a referral by the United Nations Security Council or a declaration by the relevant state allowing the Court to exercise jurisdiction.
So my understanding is that perceived crimes committed in Palestine receive automatic jurisdiction when they officially become a member of the ICC. Crimes committed outside, i.e. Israel, must be referred by the UN Security Council.