We might not be in this mess now, and it might not be with these belligerents, but we would still be in much the same mess, as it dates back to the dawn of recorded history. Any "stability" is simply a nice political way of saying "we're resting between rounds".
The short version is that the nice relatively-fertile land we now call the "Middle East" was a convenient destination for refugees and conquerors from the kingdoms to the north, east, and west. As each group settled, they called the area their home, and ignored the claims of anyone who already happened to live there, who in turn were often exiled or subjugated.
Three thousand years later, the settlers are now distinct cultures, and their tales of settling and exile have turned into legends of a traditional homeland, often mixed with religious justification. As a result, any time an opportunity arises, a traditionalist can easily stir up support for reclaiming the land for his people. Other folks like to call his cause "extremist" to trivialize his intent, and we also like to point to an arbitrary moment in history when his group wasn't in control, downplaying his claim to the area. Do note that I'm not talking about any particular group or ethnicity... their histories are all pretty much the same.
The only time there hasn't been some kind of uprising is when there was a ruler oppressive enough to keep all such rebellions under control. That tactic really started in the area with the Assyrians, until they grew too complacent, and a rebellion erupted with foreign assistance. More recently, the Soviet Union and Saddam Hussein played the role for various areas, with similar ends.
There are only two ways that I see to have lasting peace in the area. One option is to nuke the entire area completely, with fallout, denying it to everyone. The tales of exiled peoples will get worse, but nobody will really want to go home for a few thousand decades. The more reasonable option is to keep working toward "political ends", supporting whatever group says they won't try to kill their neighbors. Perhaps in a few thousand more years, the distinct cultures will merge together enough that they no longer care about conquests predating the alphabet.