Back in May, I spent two weeks discussing why only men can become priests. I still have a lot of comments to go through from that time period, so I might as well restart the series.
Another reason that only men can become priests is the fact that all priests are fathers. Indeed, the idea of priest as spiritual father is so ingrained in Catholic practice that we literally use "father" as a synonym for "priest". Also, when a man becomes a priest, the word "father" becomes part of his name. For example, if a man named Peter is made a priest, he is called "Father Peter". If he prefers to be called by his last name, he goes by "Father [Last Name]".
Of course, the word "father" implies the existence of a mother. Where the priest is a father, the mother is the Church. The priest is, in effect, married to the Church. In every language, the Church is always identified as female. Souls are always female, too, whether you're discussing a man's soul or a woman's soul.
The analogy of marrying the Church comes directly from Jesus. He applied this analogy to himself a few times, calling himself the Bridegroom (of the Church). If you're interested in this topic, you can find many resources on it, including an address by Pope John Paul II and a Wikipedia page about this. Personally speaking, I haven't studied this topic, so there's not much I can say on it.
Now, objectors would say that "priests are fathers" is merely an analogy, rather than a spiritual reality. They'll probably have a hard time proving that.
On a personal note, when I entered the seminary to become a priest, people started calling me "Brother". It took a while for me to get used to it.