According to Catholicism, why is the Holy Spirit never spoken of as the Father of Jesus?

Though not a Catholic, something @MattGutting said struck me as particularly important and pregnant (no pun intended) with meaning. He said,

"Thus, God the Father is THE Father; the Son proceeds from him, and the Spirit from both. St. Thomas Aquinas, in the Summa Theologica, gives a philosophically dense discussion of the topic in the 'Treatise on the Most Holy Trinity' (First Part of the Summa, Questions 27–43). In Question 27, Article 1, he discusses what he calls 'procession' in God. He gives the analogy of a thought, or a word, of a person. The thought remains in the person, but at the same time can be thought of as separate from the person. It is in this sort of sense that the Son "proceeds from" the Father . . .."

After this quotation, @MattGutting inserts a quotation from Aquinas's Summa Theologica which contains a key word; namely, similitude.

Similitude, from a rhetorical point of view, is a kind of intellectual argument that appeals to similarity. Similes and metaphors are the most common and noticeable building blocks (or foundations) of arguments from similitude. For a more detailed look at this concept, see Richard Weaver's book The Ethics of Rhetoric (and for a brief introduction to it, go here).

When attempting to grasp the interrelationships within the Holy Trinity, we almost of necessity have to resort to similitude, as in the simile "the Holy Spirit is like ______," or the metaphor, "The Holy Spirit is the gasoline, as it were, that runs the engine of the Father, while the Son is the actual car in motion." Obviously, this kind of statement is patently absurd, but I use it for strictly illustrative purposes.

The concept of fuel/gas, however, is perhaps quite apt vis á vis the function of Holy Spirit within the Godhead, in general, and within Jesus, in particular. Fuel, of course, is the source of power, once it is ignited. In other words, fuel is a metaphor for motivation, for example.

Motivation, it has been suggested, is the act of "throwing oneself out of order and then achieving order once again" (look here for background information on Louis Tice, founder of the Pacific Institute, to whom I am indebted for this definition of motivation). This out-of-order phenomenon is characteristic of human behavior when a person's past behavior has to give way to a new behavior because of a change in goals.

Being a flesh-and-blood human being in space and time, Jesus was not exempt from this phenomenon. When he had completed one particular aspect of his God-given ministry, he was motivated to embark on another aspect of ministry. He succeeded in each and every task precisely because he was led, filled, impelled, anointed, and empowered by the Holy Spirit at each stage in his public--and private--life. Never was there a lapse in Spirit-controlled thought, word, or deed. Jesus could say of a truth that the food which sustained him was doing the will of his Father (see John 4:34).

In the life of Jesus, then, we notice that he was "motivated" to go from one place to another at the instigation of the Holy Spirit. After Jesus' baptism by John and just prior to Jesus' 40-day solitary sojourn in the wilderness wasteland, Matthew tells us Jesus was

". . . full of the Holy Spirit, [and he] returned from the Jordan [River] and was led up by [or led around by] the Spirit into the wilderness to be tempted by the devil" (4:1 NASB Updated, my italics).

Mark adds this thought:

"Immediately [after his baptism by John] the Spirit impelled Him to go out into the wilderness" (1:12 NASB Updated, my italics)

John adds this thought:

"'I have seen the Spirit descending as a dove out of heaven, and He remained upon Him'" (1:32, my italics).

Putting these verses together, we find that the Holy Spirit of God

1. Filled Jesus

2. Led Jesus

3. Impelled Jesus

4. Remained on Jesus

These are all "action words," and for there to be an action, there needs to be an expenditure of energy. Luke tells us in 4:14,

"And Jesus returned to Galilee in the power of the Spirit, and news about Him spread through all the surrounding district" (my italics).

We can therefore add a fifth action word--power--to the list of four, above. Jesus was empowered or energized by the Holy Spirit to do what his Father wanted him to do; namely, to preach and teach the gospel, which the prophet Isaiah prophesied and which Jesus fulfilled:

"THE SPIRIT OF THE LORD IS UPON ME, BECAUSE HE ANOINTED ME TO PREACH THE GOSPEL TO THE POOR. HE HAS SENT ME TO PROCLAIM RELEASE TO THE CAPTIVES, AND RECOVERY OF SIGHT TO THE BLIND, TO SET FREE THOSE WHO ARE OPPRESSED, TO PROCLAIM THE FAVORABLE YEAR OF THE LORD" (Luke 4:18-19 NASB Updated).

The above quotation, which Jesus read from the scroll of Isaiah in the synagogue in Nazareth on the Sabbath day, gives us a sixth word about the ministry of the Holy Spirit in Jesus' life and ministry; namely, anointed. How fitting that the Holy-Spirit-anointed Son of God should be called Messiah. As Constable observed,

The name Christ is the rough equivalent of the Hebrew “Messiah” or “Anointed One.” In the Old Testament it refers generally to people anointed for a special purpose including priests, kings, the patriarchs (metaphorically), and even the pagan king Cyrus. It came to have particular reference to the King whom God would provide from David’s line who would rule over Israel and the nations eventually (cf. 2 Sam. 7:12-16; Ps. 2:2: 105:15; et al.). The early Christians believed that Jesus of Nazareth was the Christ of the Old Testament. Because they used both names together, “Christ” became a virtual name for Jesus, a titulary (title turned name). Paul, for example, used it this way frequently in his writings.

In conclusion, as you can tell by now, the Holy Spirit, the Son, and the Father have what could fairly be called distinctive roles in the unfolding of salvation history from eternity to eternity. If we expect to comprehend--in this life, at any rate--how the three persons of the Godhead function as one and yet at the same time maintain a distinctness of roles, our expectations will be dashed every time. We need, therefore, to tread lightly and reverently, with feet bereft of shoes (see Exodus 3:5) as we contemplate what is essentially ineffable.

Jesus was begotten BY God, THROUGH the power of the Holy Spirit. He was not begotten by the Holy Spirit, so the Holy Spirit is not His father. (See John 3:16)

Since God has a body, He would have had to have sex with Mary to impregnate her. But then she wouldn't have been a virgin. However, since the Holy Ghost does not have a body, he could do the job on God's behalf, without leaving a trace of penetration. (remember, ghosts can walk through walls and stuff.)

Following @FMS' comment, I've completely rethought my answer.

It helped that I discovered that (as usual) St. Thomas Aquinas was there before me. He wrote about the question in the Third Part of the Summa Theologica, Question 32, Article 3. Your question is brought up (more or less) in Objection 3 of this article:

Further, God is called our Father by reason of His having made us, according to Dt. 32:6: "Is not He thy Father, that hath possessed thee, and made thee and created thee?" But the Holy Ghost made Christ's body, as stated above. Therefore the Holy Ghost should be called Christ's Father in respect of the body fashioned by Him.

It's reasonable to think that just as Mary, in whose body Jesus was conceived, was his Mother, so the Holy Spirit, by whose power he was conceived "according to the flesh", would likewise be his father. Aquinas disagrees, however, quoting St. Augustine:

On the contrary, Augustine says (Enchiridion xl): "Christ was born of the Holy Ghost not as a Son, and of the Virgin Mary as a Son."

He explains,

The words "fatherhood," "motherhood," and "sonship," result from generation; yet not from any generation, but from that of living things, especially animals. For we do not say that fire generated is the son of the fire generating it, except, perhaps, metaphorically; we speak thus only of animals in whom generation is more perfect. Nevertheless, the word "son" is not applied to everything generated in animals, but only to that which is generated into likeness of the generator. ... And if the likeness be perfect, the sonship is perfect, whether in God or in man. But if the likeness be imperfect, the sonship is imperfect.

Now, it must be observed that what is said in its perfect sense of a thing should not be said thereof in its imperfect sense: thus, because Socrates is said to be naturally a man, in the proper sense of "man," never is he called man in the sense in which the portrait of a man is called a man, although, perhaps, he may resemble another man. Now, Christ is the Son of God in the perfect sense of sonship. Wherefore, although in His human nature He was created and justified, He ought not to be called the Son of God, either in respect of His being created or of His being justified, but only in respect of His eternal generation, by reason of which He is the Son of the Father alone. Therefore nowise should Christ be called the Son of the Holy Ghost, nor even of the whole Trinity.

In other words:

• There are multiple ways, literal and metaphorical, to use the term "Son". The closer the likeness of one being to another from which it sprang, the more literally (the more "perfectly" in Aquinas' terms) we use the term "Son".
• If we use a term literally of a person, we don't use it metaphorically. For example: We can have a picture of Socrates and say (of the picture) "This [meaning the picture] is a man". Of course, it's a picture of a man; we use the term "is a man" indirectly, meaning "This is a representation of a man". But if I have Socrates here, and I say, pointing to him "This is a man", I don't mean "This is a representation of a man", but literally "This is a man." We can use the term literally, or exactly, or (as Aquinas says) perfectly; therefore we don't use it non-literally, inexactly, or imperfectly.
• Jesus is the Son of God in an exact, literal sense, being the Second Person of the Trinity. We can use "Son of God" literally for him, and this kind of "sonship"—a kind of relationship Aquinas calls "procession"— is a more immediate and intimate form of generation than physical generation and physical sonship.
• Therefore (by the second point above) since we use the term literally or perfectly for him, with respect to his identity as "part of" the Trinity, our use of "Son of God" does not apply in a less immediate or exact way, to mean "physical offspring of God".
• Therefore, when we call Jesus "Son of God", we're not talking about the fact of his physical origin and conception. (When we call him "Son of Mary", we are. But "Son of God" has a precise Trinitarian meaning which takes precedence over the "physical offspring" meaning.)
• And therefore, it's not appropriate to call him "Son of the Holy Spirit"; when we talk about him being a "Son", we're talking solely about his "generation from" the Father as the Second Person of the Trinity.