That depends on whether those elected to power are islamic hardliners or not. Just because a government is duly elected by voters doesn't mean islam will just fade away.
Don't expect too much. Look at Venezuela and Columbia, which also have the forms of democracy. The former is led by a typical caudillo; the latter has a decent government but cannot maintain order in the country. In the Middle East, Pakistan and Yeman have some of the forms of a civil society, but the real power is exercised by tribal rulers. Hopefully, the Iraqis will get their acts together.