It would open the flood gates on litigation regarding matters that are considered religious in nature. For example, sex discrimination in allowing females to hold certain positions such as priests in the Catholic Church or women Rabbis. Additionally, there would circumstances where ethnic discrimination would be tried such as: Orthodox Jews only consider someone Jewish if their mother was Jewish even if the person considers himself Jewish and has practiced as such their entire life.
The First Amendment provides protects Religion from being State controlled. "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof..." (see 1st Amendment) The EEOC was created by Congress in passing the Civil Rights Act of 1964.