The list changes over the years but right now:
Stan Rogers - Louise’s Song
John Eddie - It Doesn’t Get Better Than This
(starting around :40) John Eddie - It Doesn't Get Better Than This - 2/28/10 - YouTube
Yours?
Stan Rogers was awesome – an unacknowoedged hero.
I’ll nominate two:
Old school: “I Only Have Eyes for You”
New style:
“You’re the Reason God Made Oklahoma”
The First Time Ever I Saw Your Face - Roberta Flack’s version.
Christina Perri, “Arms.”
That reminds me of another one, since Priscilla Herdman did a version of “The First Time…” – her “Music to Me”.
jtur88 – yes, I love Stan and it makes me happy and sad to listen to him. “Lies” is my all-time favorite but didn’t exactly qualify as a love song.
George Harrison’s
Something (Pattie Boyd, later known as Layla).
Even Frank Sinatra called it the “greatest love song ever written” (or so I heard).
“Here, There and Everywhere” by the Beatles, primarily Paul McCartney. Sure, you could say it’s a bit . . . clingy, but God that melody’s gorgeous.
Also, “Hey, That’s No Way to Say Goodbye” is one of my all time favorite songs. It’s both a love song and melancholy - sweetness, but with an undercurrent of loss. Not really the type of thing for weddings, but I think it qualifies.
Lastly, for something a bit more recent, “I’ll Follow You Into the Dark” by Death Cab for Cutie (can you tell I like acoustic guitar yet?). Some might say it’s a bit too cute, and maybe it is, but it’s absolutely gorgeous. Plus, the music video’s great.
ETA: OK, the real last one: Iron and Wine’s “Naked As We Came”. Again, not wedding material, but the lyrics are otherworldly. One will spread our ashes 'round the yard . . .
Johnny Cash’s version.
“In Spite of Ourselves,” John Prine and Iris DeMent
“Volverás”, by Ricky Martin.
I also like “Never My Love” by The Association, and “Heart Like a Wheel,” written by the MacGarrigal sisters but sung by Linda Ronstadt.
At least two of these three songs are bittersweet. Actually, all three of them could be considered thus.
Ah, but the Ronstadt version I have leaves off the best verse
Some say death is a tragedy
It comes once and it’s over
But my only wish is for that deep dark abyss
For what’s the use of living with no true lover?
Good point. Unfortunately, in my mind I can only hear that sung in Kate and Ann’s horrendous hillbilly warble. 
Speaking of Kate and hillbilly warbles, Emmylou’s Harris’s tribute to Kate is a friendly love song. Lyrics for Darlin' Kate by Emmylou Harris - Songfacts
Al Green, Let’s Stay Together
From George Harrison’s All Things Must Pass album, “I’d Have You Anytime” and his cover of Dylan’s “If Not For You.”
I’ve got to give a nod to “Maybe I’m Amazed” by Paul McCartney.
Still - Lionel Ritchie
Lady in Red - Chris DeBurgh
Poison and Wine - Civil Wars
If I Needed You - Townes Van Zandt
“Stardust,” as sung by Nat King Cole
“Shining Star,” as sung by the Manhattans
“Top of the World,” as sung by the Carpenters
Alan Jackson, Remember When
Jackie Evancho Your Love
Yes, I love that song… but that video, wish I could scrub my brain of it. Nearly ruins the song for me.
I love, love, LOVE, To Love Somebody by the Bee Gees. I’d say it’s the perfect love song if it weren’t for the line:
*In my brain,
I see your face again, *
Well, guess you gotta make it rhyme.