To understand the answer provided below, we need to reframe the original question slightly. Perhaps a better way to put "was libration first predicted" is to say, outright, libration was first observed, then a theory was advanced explaining its occurrence. Although the observation of libration is documented in 1632 by Galileo, and independently observed and documented in 1647 by Hevelius, the question opens up an initially unseen view of history regarding who first developed a theory for libration. Reading relevant portions of Newton's preserved correspondence, clearly Newton was working in depth on the libration problem by about 1670, or so. From about 1670 and, more or less, the decade following, the matter of lunar libration was the subject of discussion among Newton and Huygens, as well as with Halley, and Mercator, among others. The history of events shows that to no small degree, in later life Newton was trying to pin down the primacy of his work regarding libration. There seems to have been an air of intrigue regarding when libration had been first explained, others were trying to claim they were first in line. The general view of the Ptolemeo-Aristotelian cosmos is changed by Newton's presentations in Principia Mathematica, and regarding libration, Newton wanted a solid and honest cornerstone establishing the date of his work as being the first. But apparently there is no extant evidence that lunar libration had been predicted and then confirmed by observation.
Alan Gabby discusses the discovery of lunar libration by Hevelius and development of the theory regarding it by Newton, in an essay titled Innovation and Continuity in the History of Astronomy: The Case of the Rotating Moon. This essay is within a larger anthology, Studies in Philosophy and the History of Philosophy, Volume 24, Revolution and Continuity: Essays in the History and Philosophy of Early Modern Science, edited by Peter Barker and Roger Ariew. The publisher is Catholic University Press (1991).
Gabby gives the following account on page 96 and following pages:
Galileo gave his account of parallactic lunar in the Dialogo of 1632, and in correspondence of 1637-38 (albeit probably unknown to Newton) announced further parallactic libratory effects. Libration in latitude and longitude was discovered by Hevelius, who produced the first maps of libration in Selenographia (1647). In 1648 Hevelius hit on the idea of explaining longitudinal libration by having the Moon face always its eccentric point, not the Earth. This idea occurred independently to Riccioli, who published it in Almagestum novum (1651), together with libration maps and a general survey of the subject. However, Riccioli was unhappy with the eccentric hypothesis, since the observed librations were not quite those predicted from the accepted value for lunar eccentricity, a weakness that Hevelius recognized, though he favored retaining the hypothesis until something better came along.
Gabby gives a detailed account of Newton's involvement in the development of an understanding of lunar libration and its occurrence. Mercator, in his Institutionum astronomicarum libri duo, (1676) provides the earliest version of Newton's libration theory as told to him by Newton, both in person (presumably in discussions with Mercator at Royal Society meetings) and later by letter. Newton apparently had ready a text for publication on the subject in 1673. As Gabby states,
Newton’s letter to Oldenburg for Huygens of 23 June 1673...strongly suggests that the theory itself was in existence by that date at the latest.
Clearly, Newton had the theory at hand yet before mid-1673, but details of the time are unavailable. However, Newton's own recollection of the inception of the theory was documented by Newton some 40 years later in the third edition of his Principia Mathematica. Those details come, somewhat, from a draft of that edition indicating Newton may have been uncertain, precisely, of the date. Without his preserved correspondence, and that of others, as well as the communications with, and publication by, Mercator, those details possibly would have remained unclear. His notation in the draft was for deletion of "anno 1675" regarding the date of which he had given Mercator details of the libration theory. Gabby takes particular note,
Whether Newton’s deletion of “anno 1675” signifies an uncertain recollection of the date of something that took place at least forty years previously, or a decision to mention only the date of publication, it seems safe to link Newton’s lunar libration theory, his first published contribution to astronomy, with the years 1673—75.
But did Newton have another reason for establishing this date for the primacy of his work? Why would Newton, now in his old age, be so insistent on the veracity of this date? Gabby makes the following comment:
What prompted these persistent efforts in Newton’s old age to get right for first publication the wording of what we now realize is a priority claim? It seems to me that Newton had read or heard about (I cannot say when) the theory of lunar libration in longitude that [Giovanni Domenico] Cassini had published in 1693. Cassini’s theory seems to date also from circa 1675 and also employed, though none too clearly, the model of a moon rotating uniformly about an axis inclined at 7.5° to the orbital axis (and at 2.5° to the ecliptic axis), and moving nonuniformly in its orbit. No doubt it was G. D. Cassini (whose striking contributions to the observational study of ultra-lunar rotations Newton recognized and used) who was the implied foreign rival claimant to the libration of the moon to whom Newton wished to deny priority in the third edition of the Principia...
This was a very interesting question. Nevertheless, things are not always as they seem... One cannot prove the negative, but apparently there is no record of a prediction of lunar libration, and then an observation to confirm it. Rather, first lunar libration was observed, and then a theory for its occurrence was advanced.