Today (YouTube), by The Jefferson Airplane gets my vote.
Wow! These are some fantastic suggestions! iTunes thanks you all for having me buy so many of these songs.
So far “Darling, Be Home Soon” captures best what I’m after.
Keep the suggestions coming! Let me add to the criteria any song that personally moved you in the sense of just being glad, satisfied, honoured, etc., that the other person is a part of your life.
Bonus points if it contains a phrase in French (“Ne Me Quitte Pas” is a beautiful song that just doesn’t quit align with the emotion I’m seeking).
The greatest love song ever written is “Every Time We Say Goodbye” by Cole Porter. Any version. That’s how good the song is. And it’s the work of a genius. If you doubt me, try writing a classic, timeless tune, a melody that everyone acknowleges as a great melody, where no less than the first eight notes are exactly the same.
If you want something slightly more contemporary, albeit still over 25 years old, I’d suggest “Songbird”, a song usually credited to ‘Fleetwood Mac’ (‘Rumours’, of course) but it’s really a solo Christine McVie piece and should be credited as such. She wrote it, she sang it, she played the solo piano. And it’s very, very hard to beat.
"Ribbons In The Sky" is Stevie Wonder’s finest ballad, from the period when he had talent to burn. Another song that dispenses with all the rules, and becomes just a free form expression of intense, almost inexpressible love. I love it because it’s infused with affection and respectful romance. The singer is being polite, and asking if he may hold her hand.
"To Make You Feel My Love" is another great song. Any version except Dylan’s will do.
I remember a song by an 80’s band Translator called "I Love You, off the album “No Time Like Now”… can’t find the lyrics online, but ordered the CD for $9 though (woo hoo!!!)
Here are two beautiful songs. I thought of these immediately so I’m including both, although the second may be more of what you wanted.
I have often walked down this street before;
But the pavement always stayed beneath my feet before.
All at once am I
Several stories high.
Knowing I’m on the street where you live.
I love this song. Gorgeous melody and wonderful lyrics. **On The Street Where You Live ** from My Fair Lady.
From Miss Saigon: The Last Night of the World
*Dreams
they were all I ever knew
Dreams
you won’t need when I’m through
Anywhere
we may be
I will sing with you
a song…
A song played on a solo saxophone
So stay with me
and hold me tight
and dance
like it’s the last night of the world *
Forgive the self-indulgence.
I appreciate this song doesn’t quite fit the zeitgeist of your thread but I’ve been listening to a lot of Scott Walker recently, both with and without the Walker Brothers. (Scott’s version of If You Go Away is superior to Brel’s IMO but I can’t locate a video of it.) Regular listeners to Walker will know that a large box of Kleenex For Men is a prerequisite for most of his work and that counselling may be necessary if an entire album is attempted in one sitting.
When I’ve stopped crying I may commence a thread about torch songs (with optional orchestra).
Good topic.
What is this for?
A mix tape for the one you love? ** Nick Cave’s** Into My Arms is beautiful. Not so much of the sexual tension, but there you go.
A post break-up wallow? Hmm…what about Jeff Buckley’s Lover, You Should Have Come Over.
It’s a song about wanting badly to be forgiven by someone you’ve cheated on so that you can have hot, monkey sex with them, and it’s pretty great.
*"Lonely is the room, the bed is made, the open window lets the rain in
Burning in the corner is the only one who dreams he had you with him
My body turns and yearns for a sleep that will never come
It’s never over, my kingdom for a kiss upon her shoulder
It’s never over, all my riches for her smiles when I slept so soft against her
It’s never over, all my blood for the sweetness of her laughter
It’s never over, she’s the tear that hangs inside my soul forever"*
or how about Joni Mitchell’s A Case Of You?
A soundtrack for the bedroom?
I like Pulp’s F.E.E.L.I.N.G.CA.L.L.E.D.L.O.V.E. It’s sleazier, definitely no syrup, definitely sexual and mildly explicit.
Ok, I have horrible taste in music but there you go.
The Righteous Brothers’ Unchained Melody
and the Beatles’ Here, There and Everywhere.
My all time fave for a romanitc song is I’ll Be There by The Jackson 5. Unless that’s too “syrupy” for you, in which case I say, you can’t wrap your sausage in the pancake without some sweet, sweet syrup! Here’s a clip of a live performance on YouTube (kinda wobbly sound quality though).
Sticking closer to the criteria in the OP though, I would suggest U2’s song With Or Without You.
Or on a completely different (hip-hop) tack, Method Man and Mary J. Blige’s duet You’re All I Need (I’ll Be There For You).
(I never realized how many music video/live performance clips there were on YouTube; it ain’t all just cats crapping on the john any more, eh!)
All That Matters by Mark Knopfler. Its kinda old-timey, if that’s your thing.
I could also make a pretty strong argument for That’s No Way to Say Goodbye by Leonard Cohen.
Pistols at dawn and wigs on the green, my good man. Here’s why I think so.
Personally, I don’t find him that arousing. 
At Last
At last, my love has come along
My lonely days are over
And life is like a song
Oh, yeah, at last
The skies above are blue
My heart was wrapped up in clovers
The night I looked at you
I found a dream that I could speak to
A dream that I can call my own
I found a thrill to rest my cheek to
A thrill that I have never known
Oh, yeah when you smile, you smile
Oh, and then the spell was cast
And here we are in heaven
For you are mine
At last
Get the version by Jimmy Scott, which would melt carbon steel.
While not very sexual, two of my favorite love songs are I’ll Catch You by Getup Kids and Little Moments by Brad Paisley, which I feel captures the feeling of being in love really well with this verse:
*When she’s layin’ on my shoulder on the sofa in the dark
And about the time she falls asleep so does my right arm
And I want so bad to move it 'cause it’s tingling and it’s numb
But she looks so much like an angel that I don’t wanna wake her up
Yeah I live for little moments
When she steals my heart again and doesn’t even know it
*
The problem I have with Brel’s Ne Me Quitte Pas is that he is too needy for my taste. I prefer people who are being dumped by other people to be a little more stoic about it, and to articulate their emotions in a more circumspect way. If You Go Away does this for me because the guy knows his woman is going to leave him and the title change isn’t a proposition, rather it is a done deal. The song may have been covered by a few less than stellar performers, as you suggest, but let that take nothing away from Walker’s rendition.
I also disagree with you on the pistols because it’s your challenge and therefore my choice of weapons. After consulting with my seconds I select stereo systems at 20 paces and may the best song win. 
“With Your Love” by Jefferson Airplane may have de-volved to the point of cliche for some, but I just love it. Hard to describe, but it just feels so full to me.
I could say the same about “Miracles.”
Wonderful Tonight by Eric Clapton.
Really just the story of a couple going out for the night, coming home with a headache and going to sleep. But it positively drips with the love the man has for his wife/girlfriend.
Something by the Beatles. Written by George Harrison, Frank Sinatra thought it was the greatest love song ever written.
FWIW, “Always With Me, Always With You” By Joe Satriani
I think I’m the only one who thinks this.
Yeah, I was thinking the same thing. Me n’ Frankie, peas in a pod! 