We already have. But will not in this political climate.
Thomas Jefferson: Deist at best, possibly atheist. Very "irreligious."
Andrew Jackson: never known to attend a church after the age of 10, not a member of any church nor an attendee while president. Joined a church (evidence suggests to appease his wife) a year after leaving the presidency.
Abraham Lincoln: described by close friends as an atheist. Very "irreligious."
John Tyler: "religious views are doubtful." Never joined any church, fervent supporter of separation of church and state.
William Howard Taft: before becoming president, Taft was offered the presidency of Yale University, then affiliated with the Congregational church. He turned it down, saying in a letter, "I do not believe in the divinity of christ."