Pre-Christian Greek literature shows that we have been misogynic for a long, long time, which may be the real reason women have traditionally been kept out of the priesthood. But other excuses have been made up, especially when corroboration can be found in the Bible, which used to say quite clearly that women were not worthy for the priesthood. But that was in the past; the Bible doesn't say that now. (More on this below.)
God is often referred to in male terms and angels are invariably 'male'. The first Christian priest was Jesus, and as far as we know, he never ordained women.
He could have done. His own Mother, who witnessed his first miracle and stood with him as he suffered on the cross, or Mary Magdalene, or the women of Bethany. They were all very suitable candidates but all of Jesus' disciples were men.
A Christian accepts that Jesus doesn't make mistakes. A Christian does not question whether Jesus was right or wrong to omit women from the priesthood (see 1 Tim. 2). A non-Christian of course can question this, and can speculate that since practically all the pagan religions 2,000 years ago were led by priestesses, it was important for Christianity to distance itself from Paganism, as the Jews had done.
So let's think back to that time and ask:
Well of course, they can, and there have been female rabbis outside the Orthodox community since 1972.1
In Judaism, men and women have separate roles. A woman's obligations and responsibilities are different from those of a man, but no less important. When Judaism was founded, many pagan religious ceremonies included raunchy sexual activities. Separation of the sexes in Judaism was seen as a way to help focus more on spiritual concerns, a practice also adopted by both Christians and Muslims.
Women have held respected positions in Judaism since biblical times and the male-only rabbinate stems from the primary role of a woman being seen as wife and mother. Women have been discouraged from pursuing higher education or religious pursuits, primarily because it might interfere with their role as wives and mothers.
There have been traditional considerations too. Women were omitted from classical smicha - the 'laying of hands' passed down to elders and sages by the first 'rabbi', Moses, and conferred only to men (see Num. 27:15-23). This smicha succession ended in around the 5th century and was replaced by various versions in which piety was assumed but not actually tested. Today, a rabbi is not only tested fit to give guidance on matters of Jewish Law and Responsa, but also tested in areas such as sociology, Jewish philosophy, and pastoral care.
Again, not for the faithful to question, but it's clear that Moses wanted elders to be different from those of the Pagan religions, many of which employed priestesses. Excluding women was one way to establish a distinct and special religion. Lev. 19 and 21, for example, instruct that beards must not be trimmed. Ancient Middle Easterners illustrated their ethnicity through the facial hair style, so the Jewish custom of pe'ot (side-locks) started off as an identity thing, and still is.
Since the time of the earliest humans, the male has been physically stronger to chase after the hunt whilst the mate was looking after babies. As mankind developed, became civilised and began to reason about things, it was understood that the male was also the provider of life itself, through the sperm needed to make babies. This enhanced his familial status even further, at the expense of the wife, who was there to simply to incubate the babies and then feed them. And so misogyny was firmly established.
It was not until 1827 that the existence of the female egg was discovered and only in the 20th century that ovulation was understood. Moses lived a bit before this time, and if God had mentioned these biological facts to Moses, somehow the message was not passed down the line. The understanding of the male and female roles has changed dramatically in the past two centuries.
We now know that females have ovum, confirming the fact that both male and female contribute equally in conceiving a child. Menstruation is necessary for procreation - it is not, as was assumed, an indication of uncleanliness. Neither has it anything to do with the moon. Today, clinging to the superiority/inferiority concept is silly, but secular society and the religions of Christianity, Islam, Judaism, Hinduism, etc., still harbour some old prejudices.
More enlightened secular society has responded by starting to remove sex discrimination in employment, voting rights, property ownership and many other areas. Science and education is also catching up. It wasn't until 1992 that the first female physicist received tenure at Harvard2, and to date only 3% of Nobel laureates have been women.
In Christianity, Protestants and Anglicans are moving forward but generally the Church lags. Whilst some denominations have had female ministers for many decades, the major conservative Protestants, Roman Catholics and Eastern Orthodox still say "Never!"
But in time, the whole Church will recognise that the sense of calling from God should be the prime requirement for ordination and that gender has no importance whatsoever.
Female pope? Most probably.3