This essay may not seem to have anything to do with The Lord of the Rings, or even Tolkien, but it lays the groundwork for the rest of this section of essays, so please bear with it. There's 2000 years of history involved.
One complaint many non-Catholic Christians have about Catholics is that not all of our beliefs are based on the Bible. Well, no, they're not. If you think about it historically, it's not too hard to see that this would be impossible. The Bible had to come from somewhere. With respect to what Christians call the New Testament, where it came from was the early Christian community*. At the time the New Testament was being assembled, there were lots of gospels around, even more letters, and various apocalyptic revelations. The Christian community had to sort through all of those in order to decide which should be included in Scripture and which shouldn't. In order to even be considered, the work had to be consistent with what the Christian community (a.k.a. the Church) had always believed. Note that when a replacement was chosen for Judas, a main requirement was that the new apostle be someone who had been with Jesus throughout his public ministry. That tie to the very beginning of Christian teaching was imperative in order to be sure that it was passed along correctly, especially when its transmission was still primarily oral.
That same connection to the beginning was essential when the community began deciding what should be part of written Scripture. When present-day gnostics claim that the Gospel of Judas and/or other early gnostic writings have been "suppressed," in a way they're correct. Those writings were left out of the New Testament specifically because they didn't agree with what Christians had always believed. For anyone who's wondered, this essential link is the reason nothing's been added to the New Testament since the death of the last apostle, John. The Christian faith has been passed along for 2000 years, and many people have added new insights regarding its beliefs, but nothing written since the loss of that last earthly connection to the teachings of the earthly Jesus has been given the label of Scripture.
The conundrum is obvious. If Scripture is based on what Christians have always believed, how can what Christians have always believed be based on Scripture? The two things, properly understood, will always be consistent with each other. But there's no historical reason that everything Christians have always believed would have necessarily been included in the writings that were assembled into the Christian Bible - just that nothing that disagreed with them would have been included.
So what do you do with the things Christians have always believed that didn't become part of Scripture? Catholics, as you might guess, have a word for this: Tradition (with a capital "T"). This is a very specific use of the word, not to be confused with tradition, which Catholics have (at least!) as much trouble sorting out as anyone else. True Tradition has to be traceable back to the very beginning of Christianity and includes such "problematic" Catholic beliefs as praying for the dead, infant baptism, and the primacy of the Bishop of Rome (beginning with Peter). When challenged, many Catholics can point to Scripture passages that support such beliefs (because, by definition, Tradition can't be inconsistent with Scripture), but not to Scripture passages on which the beliefs are actually based, because the beliefs existed first. Catholics, unless they've been influenced by more fundamentalist Christians, don't generally use "proof texts," because, ultimately, Scripture doesn't prove the truth of beliefs - belief proves the truth of Scripture.
Proof texts are also notoriously prone to paradoxes; you can often find statements that support "both sides of the story" in Scripture. Because of his own proclivity for paradoxes, this is also true of Tolkien's writings; just watch or listen some time as Tolkien readers throw quotes back and forth supporting their own interpretations of a particular passage.
Okay, so what else does this have to do with Tolkien? Primarily, I think, it means that Catholics have a somewhat different view of inspired writing than do many other Christians. For one thing, as discussed above, we don't expect the Bible to contain everything, and are open to seeing inspiration in other places. And since, you might say, we saw the sausage being made, we're perhaps - in general - more inclined than some Christians to think of the human side of the divine/human partnership needed to get God's message in human words onto the page. The New Testament wasn't delivered on golden tablets by an angel; we had to hash it out ourselves, although in faith and hope we believe it was done with the guidance of the Holy Spirit, through the mystery of divine/human cooperation known as inspiration. Saint Paul wasn't intending to provide a book for the Bible when he wrote in exasperation to those "stupid Galatians," and he didn't watch God move his quill, but the Holy Spirit guided him as he wrote - for the Galatians' good, and ours.
Could this same guidance be active (to an infinitely lesser degree) in other writings? The world at large talks pretty freely about someone delivering an "inspired" speech or writing an "inspired" book, generally meaning that it moves other people to agree with the speaker or writer. But a lot of us have also experienced, for example, a teaching or a worship service that could carry the true definition of inspiration (based on the same root as "respiration"): something breathed into us by the Spirit of God. Breathed through imperfect and limited human beings, but still carrying enough of the breath of the Holy Spirit that those of us on the receiving end can inhale it ourselves. As I hope is obvious by now, I believe we can sense its fragrance in the air of Middle-earth at the end of the Third Age. From his letters, it seems that Tolkien gradually came to believe that, too - not about all of his fiction, but about LotR.
A word on priorities:
Some evangelical Christian commentaries on Tolkien express the fear that if Christians are involved in reading his fiction they'll spend less time with Scripture than they would otherwise. Neither I nor Tolkien would consider this to be any kind of a positive development. Even though I've said that I consider LotR to be an inspired work, I've also said this is true at an infinitely lesser degree than is true of Scripture. I haven't done a count, but my guess is that in this collection of essays the number of references to Scripture is about the same as the number of references to Tolkien's fiction. And if you read JRRT's published letters, especially letters to his children, you'll find that he was no stranger to the Bible. If The Lord of the Rings touches you spiritually in a way you can't quite put into words, don't stop reading it - but find something in your life other than Scripture reading or prayer to sacrifice for the time you spend with it.
My interest in Tolkien's fiction is centered on LotR, because it nourishes me spiritually in a way his other fiction doesn't; as I've said elsewhere, I believe this is because it communicates his faith in a unique way - something Tolkien himself recognized. This approach might not be as useful for other readers. As with any kind of spiritual reading outside the Bible, spend time with what nourishes you and not with what doesn't. Just keep in mind that nourishment can be challenging as well as consoling, and can bring questions as well as answers. I'd add that one sign of good spiritual nourishment is whether the reading enriches your appreciation of Scripture when you return to it, something LotR has consistently done for me over the years.
*The early Christian community had to make this same decision about what to include in the assemblage we now call the Old Testament. This wasn't easy, as shown by the fact that 2000 years later Catholic Bibles include in "regular" Scripture some things that Protestant Bibles include, if at all, as "deuterocanonical" or even "apocryphal." Whether to include these books and parts-of-books in the Bible was a cause for heated debate when the early Church originally looked at the question, and it hasn't been resolved to everyone's satisfaction yet. (The Protestant version includes in the canonical Old Testament only those writings that had been in the Hebrew Scriptures. The Catholic version includes some sections that were originally written in Greek, so are more "modern" than the rest of the Old Testament.)
As a side note, some Catholic beliefs, such as praying for the dead, are supported not only by Tradition but also by the writings that are included in the Catholic version of the Old Testament but not in the Protestant version. So the question of whether a specific belief is found in "Scripture" can have different answers depending on whose Scripture you're looking at. The primary Scriptural support for the practice of praying for the dead, for example, is in Maccabees.