AMSTERDAM/JERUSALEM, April 3 (Reuters) - The International Criminal Court rejected on Tuesday a request by the Palestinian Authority to investigate alleged crimes committed in the Palestinian Territories, saying it had no jurisdiction in what is not a full U.N. member state.

The decision dashed Palestinian hopes that the war crimes court, based in The Hague, might examine the events of the 2008-2009 Gaza war in which about 1,400 Palestinians and 13 Israelis were killed.

The prosecutor's office said in a statement that even though Palestine has been recognized as a state by more than 130 governments, the fact that its current status before the United Nations General Assembly is that of observer, rather than a full member, puts it outside the court's jurisdiction.

Today's decision appears to close the door for now on access to the ICC for victims of international crimes committed in the Palestinian Territories - at least until the General Assembly recognizes Palestinian statehood, said Kenneth Roth, executive director of Human Rights Watch.

It also shines a spotlight on an ugly reality - many parts of the world remain beyond the ICC's reach. People from influential states and their allies have been able to evade justice for crimes in Gaza, as well as in Syria, Chechnya, Sri Lanka, Iraq and elsewhere.

Israeli Foreign Ministry spokesman Yigal Palmor said Israel welcomes the decision on the lack of ICC jurisdiction but that it had reservations about some of the legal pronouncements and assumptions in the prosecutor's statement. (Reporting by Sara Webb in Amsterdam and Crispian Balmer in Jerusalem; Editing by Robin Pomeroy)