ISTANBUL  Parliament took a major step on Saturday toward lifting a ban against women’s head scarves at universities, setting the stage for a final showdown with Turkey’s secular elite over where Islam fits in the building of an open society.

Lawmakers voted overwhelmingly in favor of a measure supported by Prime Minister Recep Tayyip Erdogan to change the Constitution in a way they say will guarantee all citizens the right to go to college regardless of how they dress.

The authorities imposed the ban in the late 1990s, arguing that the growing number of covered women in colleges threatened secularism, one of the founding principles of modern Turkey.

Secular opposition lawmakers voted against the change, with about a fifth of all ballots cast. Crowds of secular Turks backed them on the streets of the capital, Ankara, chanting that secularism  and women’s right to resist being forced to wear head scarves by an increasingly conservative society  was under threat.