ISLAMABAD, Pakistan—The difficulties confronting a group of Pakistani lawmakers in trying to shut down a religious arm of the state illustrate the struggle the country faces in determining Islam’s place in society.

The human-rights committee of Pakistan’s Senate says the Council of Islamic Ideology—a constitutional body created in 1962 to ensure Pakistan’s laws stay in line with the strictures of Islam—has outlived its usefulness and that its pronouncements hurt women and religious minorities.

...