Separation of church and state in Islam is a yes and no answer.
Firstly, there is no concept of "church" in Islam, so in that sense there is no separation of religion from the state. Ideologically there is not, but in practice there often is. This is evident by 50% of nations with majority Islamic population being democracies (or close enough to it). Indonesia, the largest Muslim nation, is evident of that - a secular democratic government with a large muslim population. Actually it is incorrect to say Indonesia is a Muslim nation. It is a secular multi-faith nation with a large number of Muslims.
Karen Armstrong, Instructor at Leo Baeck College for the Study of Judaism, in a Oct. 22, 2001 Salon.com interview titled "Fundamental Problems," stated the following:
"Even though ideologically [in Islam] there can be no separation between church and state, both Sunnis and Shiites developed a separation very early on. In the Sunni world, the separation was de facto; Islamic law developed as kind of a counterculture to the aristocratic courts. In the Shiite world, there was a separation of church and state on principle. It was held that since every state was corrupt, clerics should take no part in them, that the religious should withdraw until the messiah came and established a proper Muslim state."
Can read the 'yes' and 'no' side here:
http://israelipalestinian.procon.org...stionID=000603
The First Amendment states that "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof."
The Quran also teaches freedom of religion:
“There is no compulsion in religion — the right way is indeed clearly distinct from error.”— 2:256
Many Muslim clerics believe like Witness Lee did - they should stay out of government so they don't "defile" themselves.
Most Muslims have no interest in seeing Sharia Law replace the existing Constitution. They live their lives in peace according to their beliefs and believe that one day the secular governments will be overthrown when the Messiah comes. This is not unlike most Christians who also have no interest in seeing "Christian law" (e.g. the 10 commandments) replace existing laws of government, but believe that things will change when the Messiah comes.