Democracy does not automatically lead to peace. Democracy, as it was developed, was an outgrowth of the working and landed classes' desire to protect their own interests. They saw that flighty or too-powerful kings could take their stuff away without a second thought or any owner redress, so the strong middle class banded together to form an elected government, by the people and for the people. (And I am not necessarily talk ing about the United States here, though George Washington was the first ever elected head of government in world history.)
So, as you say, when Bush talks about democracy, he is making some mistakes. He is assuming that people naturally cry out for democracy. Well, it may or may not be true, but in America, France, Britain, and most other democratic nations, they achieved their democracies through a long, bloody internal struggle against the monarch. It was self-determination that brought our democracy. It was not foisted upon us by some well-meaning foreign nation. The democracies that exist today have it because they wanted it, and are therefore well-disposed to sorting out their differences through the ballot, not the bullet.
For instance, look at the recent Canadian elections: they had problems with their head of government and they wanted a replacement. Did they rise up in a bloody junta or coup to oust their leaders? Nope, a vote of no-confidence was tallied and the government changed hands. So there were two willing participants: the elected and the electors. A bloodless, non-violent change of power. Not so in Latin America (the so-called "banana republics"), Africa, or Eastern Europe, whose corruption-ridden nations are still chafing at the recently-loosed bit of Communism. You can bet that change of government is not accompanied with peace of any kind.
So to answer your question, democracy and peace are not hand in hand. But conversely, they are also not mutually exclusive. The Iraqis are a lot closer to freedom than they were under Saddam, and a lot closer to peace than Israel, whose arch-rivals just freely elected a group of barbaric savages into power.