The more extreme critics of American strategy in Iraq demand that American forces just leave. "Bring the troops home," is the mantra. What if that were done? It's certainly possible. We withdrew a smaller force from Lebanon in 1984, four months after suffering over 250 dead from suicide bomb attacks. The Lebanese civil war went on without us for another seven years. The fifteen year civil war in Lebanon left about two percent of the population dead, and twenty percent driven from their homes. For a country the size of Iraq, that would translate to half a million dead.
We have a larger force in Iraq, so it might take six months to get everyone out. Then what happens? No one knows for sure, but there are some broad patterns of events that are likely to continue. First of all, the government of Iraq is going to try to keep going, if only to preserve control of all that oil money. The oil is what everyone is fighting over. With the Americans gone, the Sunni Arab tribal and private (warlord, terrorist, religious) militias are no longer restrained. This would probably lead to a battle for control of towns around Baghdad (the "Sunni Triangle"). These towns control the roads into Baghdad, and any government in Baghdad survives only if it can use those roads to bring in food and other goods. Some Sunnis will side with the government, which is dominated by Shia Arabs and Kurds. This is the case because over a million Sunni Arabs live in areas dominated by Shia Arabs. The Kurds don't really care about Baghdad, but they do care about Mosul and Kirkuk in the north, cities they consider "Kurdish." The Kurds have the armed and organized militias that could fight for these two cities. The fighting would resemble what went on in the Balkans during the 1990s. That is, ethnic cleansing. Non-Kurds would be driven out (with the possible exception of some small minorities who have sided with the Kurds before.) Over a million civilians (less several thousand dead) would be forced south into Sunni Arab central Iraq.
The Shia Arabs would battle for control of Baghdad, if only because several million Shia Arabs live in and around Baghdad. The solution to this would be more ethnic cleansing. The Shia Arab troops would secure roads from the south by driving out any hostile Sunni Arab populations living along the route. In the classic Arab fashion, Shia Arab military commanders would meet with the local Sunni Arab leaders along the road, and let each know that any attacks in their sectors would result in the local Sunni Arabs being driven went, into traditionally Sunni Arab territory.
The oil facilities have been under heavy attack by Sunni Arab terrorists for over a year, and the Iraqi government has built a security force that has been able to protect those facilities. This could continue, with the Iraqis hiring more foreign military professionals to help them with the defense of their biggest asset. There are now over 20,000 foreign security specialists in Iraq, and many, if not all (if not more) of these would still have jobs after American troops withdrew. As a result, there would still be thousands of "American troops" in Iraq. But these guys would be on the Iraqi government, not the American government, payroll.
Although the Sunni Arabs are outnumbered (four to one) by the Kurds and Shia Arabs, and cut off from the oil revenue, they have more military and administrative expertise. The Sunni Arabs also have the overt, or covert, aid from all the other Sunni Moslems in the world (about a billion people). Another thing the Sunni Arabs would do is give al Qaeda a new home. Al Qaeda has been openly allied with the Iraqi Sunni Arabs for over a year, and their fanatic fighters would serve the Sunni Arabs, just as they did in Afghanistan, before U.S. troops and bombers arrived. The Sunni Arabs have already pitched their struggle as a Sunni versus Shia thing, and with the Americans gone, they could be a lot more blatant about it. This aid would bring in volunteers, money and weapons. The Iraqi Shia Arabs can call on help from Iran, which would bring in volunteers, money and weapons.
It's also possible that neighboring Sunni Arab states (especially Saudi Arabia, Syria and Jordan) might send military units in to help their fellow Sunnis, but this would risk open warfare with Iran. This is something Arabs do not want, because over several thousand years of fighting Iran, the Arabs have lost nearly every time.
If the Kurds seized control of the northern oil fields, and proclaimed their independence, this would probably bring in the Turkish army. Thus the Kurds would probably stick with the fiction that their territory is part of Iraq. The Kurds would have to continue fighting, because the Sunni Arabs they drove out would be eager to regain their lost territory, and oil.
The Lebanese civil war, which was just about control of real estate by ethnic and religious factions, went on for fifteen years. A similar Iraqi war might not last that long. Then again, the Iraqi Sunni Arabs are pretty determined, and their opposition has some weaknesses. The Iraqi Shias and Kurds both suffer from factionalism. While you'd think that the pressure of war would cause the factions to cooperate, this was not the case during the Lebanese civil war. The Sunni Arab minority in Iraq has ruled the place for a long time by exploiting the factionalism among the majority. One thing the Sunni Arabs would strive for was cutting off the flow of Iraqi oil. They might or might not be able to do this, and those attempts would drive the price of oil even higher.
What would happen if American troops withdrew? Now you know, sort of. Well, at least you have a better idea. For details, study the Lebanese civil war of 1975-90, the wars in the Balkans during the 1990s, and the ongoing violence in Darfur and Congo.