The question is asked as though it had one answer. It has as many answers as there are traditions, and in several of them the answer is itself under active dispute.
Settled, in favour
Reform Judaism ordained its first woman rabbi in 1972, Reconstructionist in 1974, Conservative from 1985. Most mainline Protestant denominations ordain women to full ministry and to the episcopate where they have one. Haitian Vodou has never distinguished: the mambo and the houngan hold the same office with the same authority. Shinto ordains women as kannushi. Sikhism has no doctrinal bar to a woman serving as granthi.
Settled, against
The Catholic and Orthodox churches restrict priestly ordination to men. Rome's position was stated in Inter Insigniores in 1976 and again in Ordinatio Sacerdotalis in 1994, which declared the church has no authority to ordain women, though whether that declaration is infallible is argued by Catholic theologians and has never been resolved. Orthodox Judaism does not ordain women as rabbis. Sunni and Shia Islam do not permit a woman to lead a mixed congregation in prayer, and the classical schools agree on this while differing on women leading other women.
Contested now
- Theravada Buddhism. The bhikkhuni lineage lapsed and was revived from 1996 using an East Asian lineage. Sri Lanka has largely accepted the revival; Thailand and Myanmar have not. See the separate article on the bhikkhuni entry.
- Modern Orthodox Judaism. Women have been trained and given titles such as maharat and rabba, and appointed to congregational positions. The Rabbinical Council of America and the Israeli Chief Rabbinate have rejected the practice. Women's advanced Talmudic study, by contrast, is now widespread and largely uncontested, so the argument is about authority, not about learning.
- The Catholic diaconate. Whether women can be ordained deacons, and whether the ancient deaconess held the same order as the male deacon, has been under formal study without a conclusion. Several Orthodox churches have taken preliminary steps toward restoring the order.
- Ifa. Women initiated as iyanifa are recognised by some lineages in Nigeria and the diaspora and not by others. The disagreement is internal.
- Anglicanism. Provinces in communion with each other disagree, and the disagreement has produced parallel oversight structures inside single churches.
The historical record
The assumption that priesthood is male by nature does not survive contact with the evidence. Rome had the Vestals, with legal capacities no other Roman woman held. Athens had the priestess of Athena Polias among its most senior religious officials. Egypt had the God's Wife of Amun. Medieval abbesses held jurisdiction over parishes and appointed clergy, and sat in imperial diets. Women are named as authors of Rigvedic hymns. None of this settles a theological question, and it does dispose of the claim that there is nothing to explain.