Yes, he used the exact phrase "cosmic horror."
But not to describe the beings of which he wrote!
So far as I can make out, when he mentions cosmic horror --whether in his stories or his essays-- it is the ideas, not the monsters, to which he refers.
In his essay Supernatural Horror in Literature, Lovecraft describes his philosophy of cosmic indifference: the notion that the best we can possibly hope for is that the universe and its inhabitants don't care about us. He uses the phrase "cosmic horror" to describe a particular kind of topic found in "folklore and its consciously artistic counterparts," of which he sees himself the heir:
The Scandinavian Eddas and Sagas thunder with cosmic horror, and shake with the stark fear of Ymir and his shapeless spawn; whilst our own Anglo-Saxon Beowulf and the later Continental Nibelung tales are full of eldritch weirdness. Dante is a pioneer in the classic capture of macabre atmosphere, and in Spenser’s stately stanzas will be seen more than a few touches of fantastic terror in landscape, incident, and character.
In this essay he also describes what sets "the literature of mere physical fear and the mundanely gruesome" apart from "the literature of cosmic fear in its purest sense."
The true weird tale has something more than secret murder, bloody bones, or a sheeted form clanking chains according to rule. A certain atmosphere of breathless and unexplainable dread of outer, unknown forces must be present; and there must be a hint, expressed with a seriousness and portentousness becoming its subject, of that most terrible conception of the human brain—a malign and particular suspension or defeat of those fixed laws of Nature which are our only safeguard against the assaults of chaos and the daemons of unplumbed space.
If he intended the phrase "cosmic horror" to create ironic contrast with the chaotic beings depicted in his works, it seems that this is what he meant by it: the philosophy of his "fear-literature" is to depict a universe that will not intervene on our behalf; that is, the beautiful order of the universe offers us no protection against the chaotic forces which rampage through it.
One last note: Lovecraft doesn't use "cosmic horror" consistently to describe his own work. The phrase is more commonly used to describe the traditions in which he follows, and he uses many other similar phrases throughout the piece, of which I present an incomplete collection below:
- fear-literature, fear-studies (1 each)
- cosmic panic (2)
- cosmic terror (3)
- cosmic horror (5)
- weird literature (8)
- cosmic fear (9)
It seems that, like with so many elements of the Mythos, the phrase "cosmic horror" began with Lovecraft but was later modified (to mean the monstrous beings rather than the philosophies they embodied) and codified (at the expense of a vast wealth of alternate phrasings with subtle nuance) by his successors.