Most indications are that Bush came into office planning to invade Iraq even if 9/11 hadn't happened, but it's of course possible that he would have decided against it for some reason. (9/11 was certainly a convenient excuse that made the idea much more popular, so maybe if that hadn't happened he wouldn't have thought he could 'sell' the idea.)
The question I would ask is what role the Iraq war played in inspiring the Arab Spring. Since about 1980 the "Big Four" power players in Middle Eastern international politics have been Saudi Arabia, Iran, Israel, and Iraq. (Egypt was a big player after WWII, but I think Sadat's assassination largely sidelined them, creating the power vaccuum that the Saudis stepped into by funding the "jihadi" movement that started in Afghanistan.)
So it's possible that seeing a heavyweight like Saddam finally go down was part of what made the Tunisians, Egyptians, Syrians, etc decide that "change was in the air" and uprisings were likely to succeed.
But maybe not; maybe it was happening anyway and Bush was just jumping the gun thinking that a US invasion was going to cause democracy to break out over the whole region. (And it remains to be seen whether Iraq, Egypt, and Syria are actually going to become functional democracies, of course--none of them are really stable enough yet to say what kind of government they'll end up with.) Saddam was pretty much brought to his knees by the first Gulf War in 91. The Kurdish areas in the north were basically independent in all but name afterwards (thanks partly to the US enforcing a continuous no fly zone right up to 2003), and he barely put down a Shia uprising in the south (who expected the US forces in Kuwait to cross the border and support them, which never happened).