In a nutshell, the close relationship between the U.S. and Israel sends a signal that any violent fundamentalists in the region will have to answer to the United States. This accountability keeps the peace process from completely unraveling. It also provides a hedge against the violent expansion of religious zealots. For example, if Muslim fundamentalists were to engineer a violent coup in Egypt, they would be forced to either adhere to the peace treaty with Israel, or thrust themselves into a war that they could not possibly win. Either way, America's close relationship with Israel maintains the balance of power in the middle east. Remove our staunch support of Israel and a tide of violent fanaticism sweeps over the region, destabilizing the entire area.
This is especially true since the collapse of the Soviet Union, who had previously funneled huge amounts of resources into the region. This post-Soviet vacuum gave rise to increased religious fanaticism as the populace looked for a new enemy to train their anger and fears upon. As Muslim fanaticism continues to spread like a bacteria throughout the region, the role of Israel as a counter balance becomes more crucial than ever.
Our close relationship with Israel also presents a rapid response option. During Desert Storm, Israeli forces prevented Saddam's troops from invading Saudi Arabia and Jordan until General Schwarzkopf arrived with the Calvary. Given the inevitability that Iran will become nuclearized, we are increasingly dependent on Israel's military might to launch quick strikes against rogue states.
At bottom, Israel provides the civilized world with a foothold in a region that has resolved to destroy us. Every American should be deeply sensible about the value of this relationship.