When others criticize religion, I tend to find that the charges they level are often addressed directly in scripture itself.
1100 years before the American constitution was written, the Muslims of the Arabian peninsula, under the guidance of the prophet Mohammed, adopted a process for selecting the leader of the Ummah, or holy community. It was that he should be chosen, not by inheritance or coercive might, but by popular acclaim.
That process was upended by political leaders centered in Baghdad, who had claimed Muslim identity in order to pull the community in Mecca into their struggle for control of the caliphate. When the leaders in Mecca attempted to expose their hypocrisy, they were murdered ruthlessly.
There was a period that allowed me to attend Friday teachings at the Conejo Valley Islamic Center, and the Imam there, addressing the prejudice facing Muslims in the world today, simply observed that if Muslims followed the tenets of Islam, such prejudice would be immaterial. His exhortation was for a return to the teachings of Mohammed.
So to Ben Carson and others, I would suggest that if they returned to the history books, they would recognize in the early years of Islam a democratic struggle that was mirrored in the formation of our nation. And maybe find cause to say that they would be pleased to be led by a man that manifested Islamic tenets both in his private and public life.