For anyone who needs a refresher:
Then Morgoth […] chose one from among the whelps of the race of Draugluin; and he fed him with his own hand upon living flesh, and put his power upon him. Swiftly the wolf grew, until he could creep into no den, but lay huge and hungry before the feet of Morgoth. There the fire and anguish of hell entered into him, and he became filled with a devouring spirit, tormented, terrible, and strong. Carcharoth, the Red Maw, he is named in the tales of those days, and Anfauglir, the Jaws of Thirst. And Morgoth set him to lie unsleeping before the doors of Angband […] For the Sindarin name, Carcharoth, the putative etymology of “red maw” serves well enough, though it is problematic at one or two points. The raw material is plain enough. Sindarin car(a)n is “red”, from the Eldarin root √KARÁN; and car(a)ch is “tooth, fang”, from the root √KARAK. The final element, roth, is probably “hollow, cave” (hence, “maw”), from the root √ROD, but this is not certain. Also uncertain is where caran has gone in the final form of the name. If it was ever really there to begin with, then it seems to have left no trace. Perhaps “red” is mere folk etymology. There’s really no sign of it in the word-form itself.
The name is attested in several earlier forms, including Carchaloth, Carchamoth, and the Qenya Karkaras. These forms are given in the very early Gnomish Lexicon, contemporary with Tolkien’s first conception of the great wolf. Also in the Lexicon is an entry carna, meaning “gore, blood, especially fresh blood”, perhaps influenced by connotations of the English carnage (a word from Latin, through French, carrying the sense of the butchery of flesh).  For the Qenya name, Karkaras, used in The Book of Lost Tales, we can turn to the Qenya Lexicon, where we find karkaras(s) glossed as a “row of spikes or teeth”. 
So much for the fictive etymology — now what about an external one? I am struck by the similarity of Tolkien’s Carcharoth to the Latin carcharus “a kind of dog-fish”, itself from Greek καρχάριας “a shark”, so called because of its sharp, jagged teeth (or κάρχαρος). The scientific name of the dreaded Great White Shark is instructive here as well: Carcharodon carcharias. “Jaws of Thirst”, indeed! Now, why do I have a mental image of Beren Camlost and Matt Hooper comparing scars? :)
These forms are so close, I think we can rule out coincidence; moreover, with Tolkien’s extensive knowledge of Latin and Greek, I cannot imagine it was an accidental borrowing. Can you? In addition, κάρχαρος is thought to show reduplication of an Indo-European root √KAR, meaning “hard”, which sounds right for Tolkien’s Carcharoth. Compare to Gorgoroth, where the reduplicative form is explicitly acknowledged (and note the coincidence of the final element, roth).
Perhaps coincidental, but offering tempting overtones, is the Greek κάρκαρον “prison”. As you may recall, the literal meaning of the Sindarin Angband (where Carcharoth was bred) is “iron prison”. The Greek word, κάρκαρον — whence Latin carcer “prison”, whence Modern English incarcerate — is said to be “of uncertain origin” by Skeat, but might it not be related to the same IE root √KAR? (Other possibilities are advanced in other etymological dictionaries, but the ones I’ve checked so far aren’t any more convincing.)
 Tolkien, J.R.R. The Silmarillion. “Of Beren and Lúthien.”
 Tolkien, J.R.R. “The Gnomish Lexicon.” Parma Eldalamberon 11 (1997), p. 25. See also Chris Gilson’s discussion of Carcharoth in “Essence of Elvish: The Basic Vocabulary of Quenya.” Tolkien Studies 6 (2009): 213–39, p. 218.
 Tolkien, J.R.R. “The Qenya Lexicon.” Parma Eldalamberon 12 (1998), p. 49.