Objection 1. It would seem that no other Divine Person could have assumed human nature except the Person of the Son. For by this assumption it has been brought about that God is the Son of Man. But it was not becoming that either the Father or the Holy Ghost should be said to be a Son; for this would tend to the confusion of the Divine Persons. Therefore the Father and Holy Ghost could not have assumed flesh.
Objection 2. Further, by the Divine Incarnation men have come into possession of the adoption of sons, according to Romans 8:15: "For you have not received the spirit of bondage again in fear, but the spirit of adoption of sons." But sonship by adoption is a participated likeness of natural sonship which does not belong to the Father nor the Holy Ghost; hence it is said (Romans 8:29): "For whom He foreknew He also predestinated to be made conformable to the image of His Son." Therefore it seems that no other Person except the Person of the Son could have become incarnate.
Objection 3. Further, the Son is said to be sent and to be begotten by the temporal nativity, inasmuch as He became incarnate. But it does not belong to the Father to be sent, for He is innascible, as was said above (I, 32, 3; I, 43, 4). Therefore at least the Person of the Father cannot become incarnate.
On the contrary, Whatever the Son can do, so can the Father and the Holy Ghost, otherwise the power of the three Persons would not be one. But the Son was able to become incarnate. Therefore the Father and the Holy Ghost were able to become incarnate.
I answer that, As was said above (1,2,4), assumption implies two things, viz. the act of the one assuming and the term of the assumption. Now the principle of the act is the Divine power, and the term is a Person. But the Divine power is indifferently and commonly in all the Persons. Moreover, the nature of Personality is common to all the Persons, although the personal properties are different. Now whenever a power regards several things indifferently, it can terminate its action in any of them indifferently, as is plain in rational powers, which regard opposites, and can do either of them. Therefore the Divine power could have united human nature to the Person of the Father or of the Holy Ghost, as It united it to the Person of the Son. And hence we must say that the Father or the Holy Ghost could have assumed flesh even as the Son.
Reply to Objection 1. The temporal sonship, whereby Christ is said to be the Son of Man, does not constitute His Person, as does the eternal Sonship; but is something following upon the temporal nativity. Hence, if the name of son were transferred to the Father or the Holy Ghost in this manner, there would be no confusion of the Divine Persons.
Reply to Objection 2. Adoptive sonship is a certain participation of natural sonship; but it takes place in us, by appropriation, by the Father, Who is the principle of natural sonship, and by the gift of the Holy Ghost, Who is the love of the Father and Son, according to Galatians 4:6: "God hath sent the Spirit of His Son into your hearts crying, Abba, Father." And therefore, even as by Incarnation of the Son we receive adoptive sonship in the likeness of His natural sonship, so likewise, had the Father become incarnate, we should have received adoptive sonship from Him, as from the principle of the natural sonship, and from the Holy Ghost as from the common bond of Father and Son.
Reply to Objection 3. It belongs to the Father to be innascible as to eternal birth, and the temporal birth would not destroy this. But the Son of God is said to be sent in regard to Incarnation, inasmuch as He is from another, without which Incarnation would not suffice for the nature of mission.
Source: Summa Theologica (New Advent)