The short answer is that the Church does not have a definitive teaching on the O.P.’s question: whether God can create “new” angels. However, the perennial philosophy that is the basis for the Church’s theological reflections suggests that, although God has the power to create as many angels as He wishes, from our point of view, the creation of the angels is already “done.” See the fuller answer below.
Man certainly does not have the power to create angels. Angels are, by nature, far superior to man. No creature can create another creature, much less a creature superior to itself.
The Church’s teaching on the angels and their creation
According to the Catholic Church, God is certainly the creator of the angels. However, there are some things to keep in mind.
First of all, the angels are pure spirits; that is, they do no have bodies, as we do, and are free from limitations that a body entails. (See Catechism of the Catholic Church [CCC] 328-330.)
Angels and their relationship to earthly time
On the level of official teaching, the Church does not address the O.P.’s question directly. However, using good theology, based on sound philosophy, one can draw some conclusions.
Time is relative to the one who experiences it
Since angels do not have bodies, as we do, their experience of time is entirely different from us.
Moreover, God is the creator of time; hence He is entirely outside of time. (See CCC 205.) He is also utterly one and simple (see CCC 202). Consequently, on God’s part there is only one act of creation, which identical with His very Essence. (See, for example, Summa theologiae [S.Th] Ia, q. 13, a. 7, 1um and ad 1um [the first objection and the answer to that objection].) That single act produces many effects—namely, us creatures—but the act as such is only one.
Therefore, it is an anthropomorphism to think of God’s creative action as occurring in time. The effects of that action take place in time, but the act as such is eternal: it is, ontologically speaking, God’s very Essence and Being.
What exactly is time?
It is useful, then, to take a moment to consider what time actually is. Aristotle defined time as “the measure of change according to before and after” (Physics IV, 11, 219b1-2) a notion fundamentally taken up by both St. Augustine (see Confessions, XI, cc. 1-28, especially chapters 24-26) and Aquinas (see In IV Phys., lc. 15-23). In other words, creatures experience time to the degree that they experience various types of changes (“movement,” in the classical terminology).
(It is interesting to note that all of our differing ways to measure time all consist in taking a particular kind of change or movement as the basis for measuring other movements. For example, we use the motion of the moon and the Sun as the basis for the length of our days and months; and other regular changes found in nature—such as the oscillations of cesium atoms—to make more precise measurements.)
How different kinds of creatures experience time
As a consequence, different kinds of creatures experience time in different ways, depending on the kinds of changes that they are capable of experiencing. Human beings, like all material creatures, undergo constant interactions with their environment. Hence, they experience time as a continuous movement.
On the other hand, angels, being pure spirits, do not experience a “continuous” time, but, if you will, “discrete” time. It depends on the changes that can take place in them: the decisions of their will, the knowledge infused into them, and so on. Since they are not material creatures, they do not have an “environment” to interact with; and indeed, their entire existence is characterized by a discrete and finite number of actions. (These actions, however, are far more perfect than our own actions, and hence although the number is finite, what they accomplish through those actions is far greater than we ever could in our lifetime. Aquinas makes an extensive treatise on the angels in qq. 50-64 of the Prima Pars of the Summa theologiae. The most relevant questions are 54 and 59, on the angelic intellect and will.)
Angelic time is simply not in the same “continuum” as our own time—it is entirely extrinsic to ours.
That does not prevent angels from interacting with us: but they are looking at our continuous time, as outsiders looking in, so to speak.
From our perspective, God has already “finished” creating the angels
What that means, from our perspective, therefore, is that God already “finished” creating the angels, even “before” He created the material universe. (In reality, “before” and “after” are very misleading here; were are talking about kinds of time that are simply incommensurable.)
Even if God wanted to create more angels—and God is quite capable of creating as many as He wants, even an infinitude, if He wishes—it would seem to us (who exist in our continuous, material time) that He has “already” created them.
That is not a defect on God’s part, as if He were “incapable” of creating angels in earthly time, but simply reflects the difference in nature between angels and men: angels, being pure spirit, are simply not bound by earthly time.
God has the power to create as many angels as He would like: even an infinitude. However, angelic nature is so different from ours (since angels are pure spirits unbound to bodies) that their experience of time is simply incommensurable with ours. Hence, it would be anthropomorphic to say that God creates new angels in our own, earthly time. From our perspective, the creation of the angels is already “done”—but this is not because God cannot create “more” of them, but because our time and their time do not overlap and is of a different nature.
As far as man creating angels, that is impossible, as only God has the power to create anything (in the strict sense—all that we humans can do is rearrange things that have already been created). See CCC 295-301 for an overview of the Church’s teaching on creation.