As a minister who performs an average of 5-7 weddings each year, I'm quite picky about the music that is allowed in the wedding service. I encourage the couples (not just the bride) to choose music that is of the highest quality and speaks of the love the Church speaks of in regards to marriage. That said -- Wagner's "Wedding Chorus" (Here Comes the Bride) is NOT allowed. The opera from which it comes does not uphold the values of marriage that the Church preaches. The couple gets married and goes off and does everything with everyone but the one to whom they are married.
I have allowed "popular" music. After listening to "I Swear" for a few times, I came to the conclusion that the love it spoke of was for all time, and not just for the moment. For that reason, I don't allow anything from "Titanic" and I also steer clear of "Phantom of the Opera" pieces. I ask couples to save Celine Dion et. al. for the reception.
Do I have the right to decide what music can be played at a wedding? You bet I do. I steer couples towards the outstanding repertoires of Bach, Handel and Beethoven. They wrote music for the Church. Bach's Mass in B minor is gorgeous. Handel's music is wonderful. It also helps that they were both Lutheran. Above all, a wedding service is a worship service. In a worship service, everything we do is to be in worship of God -- not worshipping each other. (there's lots of time on the honeymoon for that kind of stuff

) My wife came down the aisle to "Praise my Soul, the King of Heaven". Gorgeous music praising God chosen by a gorgeous bride.
I know I have disappointed many brides (admittedly, grooms rarely care about the music, if they notice it at all) with my "rules" but I also know of many more who have appreciated the guidance and direction that have made their wedding a truly unique worship service.