It depends on the parish. At the very strict parish where I attend and often play at weddings, the music director allows Pachelbel Canon. As Sarabande mentioned earlier, the Canon in D falls under the category of absolute music. There are no associations with a story, such as an opera or ballet, so I would not describe it as secular. Again, at my very strict parish, they do not allow Wagner’s Bridal Chorus or Mendelssohn’s Wedding March, or any pop music that is not religious in nature. But, the Canon is allowed. Either the sung version (with words that are religious), or purely instrumental, it is allowed.
I think what also should be mentioned is that if the “Canon in D” shouldn’t be allowed, then neither should be the Clarke “Trumpet Voluntary”, Purcell “Trumpet Tune”, Handel movements from his “Hornpipe” or “Watermusic”, Mouret “Rondeau”, Delalande “Musique Royale”, Clarke “Serenade and Fanfare”, Campra “Rigadon”, or Lully “Marche Royale” (None of these were originally written just for the organ, but were transcribed for organ for use in churches and these are often used at weddings). The other thing that shouldn’t be allowed, then would even be any other organ work composed by those such as Bach, Buxtehude, Couperin, Langlais, Dupre, Vierne, Messian, Widor, etc. Now, some of them did compose specific organ works which had titles that evoked something religious, mass-related, dedicated to the Virgin Mary, etc, but most instrumental music (organ music included) is pretty much not considered “sacred” in the sense that it has no indication that it was actually written for mass or written for something secular. That said, I know that there were specific pieces of music written for the organ for use in operas and ballets for instance.
What I think should be determined is how the music is rendered by the instrument itself. Although we could go completely “pure” and not use any instruments at all, we also do know that the church does allow for instrumental music as long as it is appropriate for use in church, such as organ works or transcriptions for organ. Also, many times feast days and other special masses such as weddings, do employ the use of strings, timpani, brass, etc. which we often see in cathedrals, at the Vatican and other churches. So, does it evoke a secular connotation when you hear it? Can the music be refined so that it doesn’t sound secular? Is it well-known enough to know that it is from a play, opera, secular oratorio, ballet, musical theater (think the organ wedding march from “Sound of Music”. Some churches have allowed this because it does sound majestic and much like you’d hear for the typical church organ - which I’m sure Rodgers and Hammerstein was going for, but it was written specifically for a musical, even though it was taking place during a Catholic wedding in the show.)
Now, I always recommend one of my favorite organ transcriptions - Marc-Antoine Charpentier’s “Prelude a Te Deum”. This was originally for orchestra used in church for Charpentier’s “Te Deum”.
I think a lot of classical instrumental music has been accepted for use at church throughout the centuries because it can ***tend ***to lend itself better for liturgy, although NOT ALL classical instrumental or religious/sacred classical vocal music would be appropriate. I think this is where people can get confused and have a hard time drawing the line for classical music at least.
For my wedding, I chose an instrumental version of Holst’s “O God Beyond All Praising” as my processional (although even though it was written for a hymn by him, the theme was still used in his Jupiter movement of “The Planets”) and we used Bach’s Organ Fugue in D-Major for our recessional (no not the Toccata and Fugue in D-
MINOR most people would be thinking.

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Here’s a thought… do we think that his famous toccata and fugue could no longer be used in church because it has become so synonomous with secular things like horror flicks, Disney’s Fantasia, etc.? It was originally absolute music, though, for use in church. I’m on the fence with that because one part of me would say yes even though it was be an incredible postlude after mass.