Most Helpful Guy
Voted neither as, from an American perspective, it should properly be viewed as a strategic calculation, not a moral decision. That said, your characterization of it as two children fighting over a toy is an unjustified trivialization of the question.
There are serious moral issues at stake, and it has been an accepted principle of international affairs - at least since the end of World War I if not before - that national self-determination is the right of a people. It's partly the reason why the victorious Allies broke up the Austro-Hungarian Empire and it is why imperialism is in such bad repute today.
That said, international affairs are about power, not principle - except insofar as sovereign powers have the ability to assert their moral principles. In the Palestinian-Israeli case, there are conflicting nationalisms, though in the Palestinian case, the sense of national identity is somewhat weak - with Palestine divided between Palestinian nationalist, Islamist and (to some degree) pan-Arab identities.
These splits within the Palestinian community have significantly weakened the Palestinian cause, with about the only thing holding the Palestinian factions together is their ardent right to return to what they consider historic Palestine. In that connection, it should not be forgotten that the United Nations Organization divided the territory in a two state solution and it was the Palestinian Arabs, not the Israeli Jews, who rejected that compromise solution.
As to the situation now, the Palestinian cause is largely over - the media images notwithstanding. The decision by the Palestinians to support Saddam Hussein's Iraq against Kuwait in the 1990-1991 Persian Gulf War effectively ended Arab support for the Palestinian cause in fact if not in name.
Throw in the rivalry between Persian-Shiite-republican Iran and the Arab-Sunni-monarchies (as well as Egypt.) You now have a de facto alliance between Israel and the Arabs, effectively shutting out the Palestinians (who are increasingly looking to Iran.)
From the American viewpoint, that Arab-Israeli alliance is useful for hemming in not just Iran, which is small beer, but Russia - which in geostrategic terms matters more. In that connection, then, support for Israel makes sense.
The Palestine issue for the U. S. is secondary and if - as they recently states - the Palestinians do not see the U. S. as a mediator, then that is their affair and the status quo will suffice. Palestinian options are few.
Most Helpful Girl
Neither, the United States should stay out