Their turnout for the 2 elections was very high, even by American standards, so it's probably not accurate to say they don't want democracy.
But is it possible that they do not, and will not, have the infrastructure and cultural traditions to handle Democracy for a very, very long time?
Just speculating.
Not necessarily. I can see two possible situations arising in Iraq (and only two). They have democracy for a short period of time before they vote in what amounts to a theocracy, which eventually will remove democractic standards. Or a civil war breaks out where the final resolution is three separate states. Iraq is too diverse to exist as a democracy. It was nothing more than a state created after WWI.
There's democracy--the expression of individual will. And then there's something different, but related: the right to live in peace without outside interference. In the midst of their new-found status as a powerless minority, the frustrated Sunnis tolerate a Jihadist faction with Baathist support that seeks to initiate a civil war. So the Sunnis aren't entirely blameless here. The Shiites, on the other hands, are taking revenge. And why not? Not only have they been suppressed for so long prior to Coalition intervention, they have withstood suicide bombings, assassinations, etc. with a degree of noble tolerance in the past few years. And the Kurds--they're just waiting out to see whether life is so intolerable in Baghdad that they either resort to official independence, or more likely retain their autonomous status and in practice, live out a relatively independent existence.
BTW, the fact that much of the country is peaceful (which is true) doesn't mean that the country is not untenable, especially given the close correlation between ethnicity and geography. During the Civil War, the fact that the South was the only region in rebellion didn't mean that American was not in peril of being split up. When you have a sizable minority (20% or up) that tolerates the use of violent means for political ends, any country has a problem.