Women's desire to be ordained as priests is misplaced for at least two reasons. First, commonly cited is the role of the priest as a spiritual father. I always think of this when I hear the response of the Vatican that they have no authority to ordain women. Truly, in a metaphysical reality, women can no more be priests than, in a physical reality, men can be mothers. Reproductive organs are not the only differences between men and women. Tolkien illustrated this with his creation of the Vala, angelic beings in his sub-created world. In the creation story of The Silmarillion, each of these beings became manifest in the world as a man or as a woman, not because they had gender as spirits but because their nature was better expressed when incarnate as one or the other.
There is something about the nature of a man that uniquely suits him to the role of an ordained priest. I will note that God the Son became an incarnate man--not a woman. And the societal roles of women at the time, and other such contextual arguments for why He came not as a woman, hold no water for me. Who is the author of history? Who is Lord of time and eternity? And what era did He deem the "fullness of time"?
No, He chose to become man because His nature as the Second Person of the Trinity was for some reason best expressed as male.
I know that some of our saints and theologians have commented on this. No doubt it has something to do with Adam, the primordial man. Likely I have even read it myself at some point. It is from those who have already wrestled with questions like this (and are far more qualified to do so than I) that I come to the second reason that people who advocate for women's ordination are mistaken.
Whereas men take on a persona of Christ in a particular way upon ordination, women, in a sense, do not need to. They are already a type of God the Son in a way that is completely unique to them. For, in the Trinity, it is the Second Person who is the Beloved. God the Father is "father" because love--as do all things--originates with him, just as physically, "father" is the active origin. Culturally, this is widely practiced in society in courtship--a man initiates, a woman is courted. Though not carved in stone, or even necessarily the only "good" way to go about it, it is revealing. Woman receives that love and loves in return, just as the Son is loved by the Father and loves him return. Just as the Son is begotten of the Father, God from God and light from light, so Eve is bone of Adam's bone and flesh of his flesh.
This was revolutionary for me when I first heard it, and it completely wiped away any nasty suspicions and insecurities about men being more "Godlike" or "Christlike." It took the idea, which I already had from Scripture, of both male and female being the image of God, and rooted it in good soil, nourishing it so that it may bear fruit in my soul.
[I am of the opinion that genetics have some formative or dispositional effect on the personality. And I remember a statement that a campus minister spiritual director at my college made--an obvious observation perhaps, but it stuck: that Jesus, having no human father, got all of his genetic material from Mary. He had no biological male source for His maleness. I haven't worked out the implications of bringing those two ideas together, nor what it means for this idea of woman as a type of Christ, yet. It's an interesting question, one I'd like to find out more about.]