A great number of Muslims don't have an issue with a democratically elected government, even one that is a bit more secular that religious, as long as the promoters of that government and those changes bring the social liberalism with it. They don't want, won't tolerate homosexuality, pre-marital sex, etc. as an acceptable way of life. These lose liberals social standards are their biggest objection to a democratic way of governing.
Perhaps if more Americans held these same social standards, we would be a stronger country. Extreme liberalism has lead to the downfall of many past worlds leading civilizations.