The answer - a resounding "no" - can only be explained in an out-of-universe manner, and can (and should) be discussed without reference to Tolkien's religious views.
In fact, Tolkien's devout Catholic beliefs didn't prevent him from enjoying Mary Renault's story The Last of the Wine, which dealt with an Ancient Greek man and his male lover. He may have also read, and served as an unofficial editor and critic of, Renault's The Friendly Young Ladies, which was about a lesbian couple (Renault was a former student of Tolkien's). He even wrote in a letter that he had been reading two other books by Renault (both of which focus on homosexual characters), and had shortly afterwards, and purely coincidentally, received a postcard from Renault herself, which became his favorite piece of fan mail:
I was recently deeply engaged in the books of Mary Renault; especially the two about Theseus, The King Must Die, and The Bull from the Sea. A few days ago I actually received a card of appreciation from her; perhaps the piece of ‘Fan-mail’ that gives me most pleasure.
-Tolkien, Letter #294
Similarly, Tolkien enjoyed the novels of Iris Murdoch, who also frequently included gay characters in her works, and was apparently bisexual herself; he was excited when he received a bit of "fan mail" from her. Clearly, if his religious beliefs predisposed him to hold negative views towards homosexuality, he didn't let that interfere with his pleasure in reading a good book.
There is one alleged instance of Tolkien ever having commented on homosexuality, found in Humphrey Carpenter's biography of Tolkien:
"Certainly the older boys [at King Edward's School, which Tolkien attended] did have prestige in the eyes of the younger, but it was the prestige of age and achievement rather than of caste, while as to homosexuality Tolkien claimed that at nineteen he did not even know the word. Nevertheless it was into an all-male society that he now threw himself."
-Carpenter, J.R.R. Tolkien: A Biography; p. 53
Now, let's move on to social and legal considerations:
Tolkien did most of the work for The Hobbit and The Lord of the Rings in the 20's, 30's, and 40's. He was a conservative British man, a fairly well to do author and professor at a world renowned University.
At that time, homosexuality was seen as a mental illness, not a sexual orientation. Homosexual activity, then referred to as "Sodomy", was illegal in Britain until the mid-60's, as far as I know. People could be arrested (and many were, with disastrous consequences for their social relations and reputations) for engaging in homosexual relations, or for merely flirting with someone of their own gender. One famous example of a person being arrested for propositioning someone of the same gender was Beatles' manager Brian Epstein; fortunately for him, his family was wealthy enough to ensure that the charges were quietly dropped and the arrest was largely concealed from public attention (until biographies were written after his death in 1967).
In fact, there was a widespread trend, beginning in the U.S., of chemically castrating gay men. Tens of thousands of people suffered this treatment in the U.S., and many more endured the same procedures in Europe, including Alan Turing in Britain.
Simply put, homosexuality wasn't on the average person's radar, so to speak. It was considered a disease, not a legitimate sexual orientation. It was also not something to be discussed in polite company, and certainly not in popular literature which might be read by children.
The following comparison seems appalling to our modern day, equality-minded sensibilities, but it would be fair to say that Tolkien didn't mention homosexuality for the same reason he didn't mention people going to the bathroom - it was considered crude and disgusting, not fit for publication. Homosexuality was thought of as grossly unnatural, morally repugnant, and unimaginably deviant in the worst possible way. It was treated as an affliction or a willful violation of universal norms of behavior and morality.
It simply would not have occurred to Tolkien to write about such things in his books. It wasn't something that "respectable" people discussed, or even thought about.
Yes, Tolkien's religious views would probably have informed his opinion of homosexuality if he ever gave any thought to the matter (which he probably didn't do very often, if at all), but the more important factor was the social environment in which he was living. It was not a subject that people were talking about, or even thinking about, back then.
I would imagine that, had Tolkien mentioned the topic in his books, he would have had a very hard time trying to find a publisher. It was illegal to publish material that might be interpreted as "promoting sodomy" - such a book would likely be deemed pornographic in nature, according to the Hicklin Test, which was formal law in Britain until 1959.
*Note: Renault and Murdoch published most of their works a good deal later than Tolkien was writing The Hobbit and The Lord of the Rings, which allowed them to circumvent the earlier laws regarding indecent and obscene literature. Renault's tendency to set her stories in Ancient Greece probably helped her avoid legal complications as well.
From a summary of Mary Renault's comments on Tolkien and his attitude towards her work, which was pioneering in its bold (for the time) references to male and female homosexuality:
Ms. Renault and Tolkien did have a dispute over the publication of her first novel, Purposes of Love. It was very racy for its time, and had hints of male and female homosexuality. Anyway, Ms. Renault wanted to use a male pseudonym, but Tolkien strongly objected, urging her to publish under her own name, or at least a female pseudonym. Indeed, she says that Tolkien strongly encouraged all the young aspiring female writers he came into contact with to reject the trend of the time for females to write under male pseudonyms and instead use their own names.
[It bears repeating that the dispute between Tolkien and Renault had nothing to do with the content or subject matter of the book, which included homosexual themes, but rather, the dispute consisted of Tolkien urging Renault to publish the book under her own name, rather than a pseudonym.]
As for Tolkien, from Letter 294:
"There are exceptions. I have read all that E. R. Eddison wrote, in spite of his peculiarly bad nomenclature and personal philosophy. I was greatly taken by the book that was (I believe) the runner-up when The L. R. was given the Fantasy Award, 'Death of Grass'. I enjoy the S.F. of Isaac Azimov. Above these, I was recently deeply engaged in the books of Mary Renault; especially the two about Theseus, The King Must Die, and The Bull from the Sea. A few days ago I actually received a card of appreciation from her; perhaps the piece of ‘Fan-mail’ that gives me most pleasure."
Note that like most of Renault’s novels, The King Must Die and The Bull From the Sea dealt sympathetically with male and female homosexual characters.
Ms. Renault died in 1983, renowned as one of the 20th century's greatest authors of gay literature.
From an article on Tolkien, mentioning his correspondence with Iris Murdoch:
Iris Murdoch, interestingly, was a tremendous fan, and loved talking to the old professor about the more abstruse points of elvish lore. When her husband John Bayley exclaimed that The Lord of the Rings was "fantastically badly written" she would look astounded, and say that she did not know what he meant.
Further information about laws regarding homosexuality in Britain, and publication of "indecent material":
The list of notable cases on this Wikipedia page includes a ban on publishing James Joyce's Ulysses.
See also this blog entry about Elven sexuality.