One of three ways, with infinite variations in between:
Worst) We get the *bleep* out. Iran, Syria and the Saudis duke it out for the territory. Iraq didn't exist until the Brits divvied up the Middle East after WW1. Iraq by all rights belongs to Iran, with tidbits to Turkey and Syria. If it goes to all all-out slugfest, Turkey may get sucked in. Good side of this: the fundamentalist Suni/Shia/Shitheads spend all their energy shooting each other, instead of mucking around with our buildings.
Best) We hang in long enough for the next Adminisration to get its shit together. The next Prez thumps heads and gets Iran, Syria, Turkey (which would be a feat, as Turkey wants no part of this mess), the Saudis and maybe a couple of others to sit down at an awkwardly shaped table (everyone has a side) and works something out. Iraq still runs the risk of getting divvied up between the various parties, see above.
Middlin' ground) We stay, we get somebody in the Administration to talk with Iran and Syria who doesn't look like a sulky five-year old (did you see that picture? OY!) and who is willing to discuss reality rather than what they'd like it to be, and get middle eastern leaders to realize they are in as much trouble with these fundies as we are. Actually more so, since the suni/shia/shitheads are local, and only need walk over to blow it up. Whatever "it" is popular at the time. All of us, them, us, Europe, possibly some help from that other HYOOJ Muslim country, Indonesia, go in and kick butt, declare martial law for five years, and then spend the next twenty getting an education system in place that teaches more than just the Q'uoran.
Where will it go? Who the *bleep* knows, as it could tip to really hideous really fast, at which point nuking it back to the stone age is a really good thought. The only thing keeping me from being that blood-thirsty is my cousin the archeologist would ... I can't think of a thing bad enough that she wouldn't do if we nuked all that archeology out of existance.