I honestly think it can't be stabilized as-is because Iraq was never a self-declared country, it was created by outside forces and held as such by an iron fist. It wants to be three separate countries.
The Sunni and Shi'a hate each other. The Kurds are hated by both and would like to be left alone. Some of the marsh arabs in the south would like to be left alone, too.
Now, if we let it be Kurdistan, (the Kurds like us), and Sunni and Shi'a states to the south, perhaps it might settle down. Then they'll just fight over the oil rights.
This post is going to make me very unpopular with the conservative/Bush supporting crowd in this thread. Which is ironic because I am myself an archconservative and a Bush supporter on most issues...except the war.
There is a distinct tendency by conservatives to react to any conversation about Iraq by blindly launching off into a tirade about how "we could just win this war if those cut and run Demorats would just get out of the way and support our troops and if you don't stand behind them you can stand in front of them and if the MSM would just report the truth you'd see how good Iraq really is."
Liberals are every bit as guilty, they just go off in the "Bush is an evil tyrant and he lied to get us into this war and where are the WMDs and Bush stole the election" direction.
Before I came here I was guilty of the same blind conservative tirades I just described. As with many things, somewhere closer to the center lies the truth. The "war in Iraq", with the goal of a united Iraq is probably not feasible. I quoted ManedWolf's post because it very accurately sums up the reality in Iraq. Notice the part I highlighted. That is the majority of the problem in Iraq today. Not al-Qaeda, not foreign fighters, not even Iran. It is that the Shia and the Sunni hate each other. They don't want to work together, they don't want to govern together, they don't want to live together.
An outstanding comparison I read recently was between Iraq and Yugoslavia. Tito kept Yugoslavia together by force and ruling with an iron hand. When he died, all the Serbians and Croatians and Bosnians and other ethnic groups suddenly had freedom...the freedom to kill each other. Trying to get them to live together just didn't work. But separating them into ethnically homogeneous countries has allowed them to live peacefully with people like them.
Iraq will probably end up requiring something similar. Splitting the Kurds off into Kurdistan would take care of them. They are already most of the way to to being an independent country anyway. It's my understanding that we are holding them back to keep from pissing off Turkey right now. Western Iraq would become Sunni-stan. If you pay attention to all the trumpeted security and pacification of the al-Anbar province, that's the real reason for it. Al-Anbar is mostly Sunni. They cooperated with coalition forces because they are tired of being blown up by al-Qaeda. It doesn't mean they like Shia though. Southern Iraq would become Shia-stan.
EDIT: ManedWolf's map came up while I was composing this lengthy post. It perfectly illustrates what I'm talking about.
There are two problems with this proposed plan. One being ethnically heterogeneous areas like Baghdad. Note all the striped areas in the map The other being the fact that most of the oil would be in Kurdistan and Shia-stan. Some kind of oil revenue sharing would have to be created between the three countries to keep the Sunnis happy.
All the Alec Baldwin jokes and the snide comments about the Democrats and the MSM aside, the reality of it is that Iraq is not a military problem. We could put a Bradley on every corner in Baghdad and it still wouldn't resolve the fundamental antipathy between the Sunnis and the Shia. We are spending 12 billion dollars and 100+ American soldiers lives each MONTH to try and unite people that don't want to live together. And this whole discussion is completely ignoring issues like corruption or how completely ate up the Iraq government is, particularly the Iraqi Security Forces.
We need to get our soldiers out of the job of try to fix this country. Will there be a bloodbath when we leave? Probably. Would there be a bloodbath anyway if we stay here longer? Probably. My favorite metaphor for Iraq is a spoiled, rebellious, criminal adult child. You care and you want to fix the problem, but at some point you have to let them sink or swim on their own. We are keeping the Shia and the Sunni apart enough that they can keep fighting. If we step back they will either finally get tired of killing each other or enough refugees will have moved around that groups will be separated enough that Iraq can have a chance at peace. Until then, we are just trying to fix a political/social/religious problem with military force. And it won't work.