“Can’t Help Falling in Love with You” by Elvis.
“Silent Night”
“Greensleeves”
“Morning Has Broken” by Cat Stevens
Same album (CD), just a few tracks later:
“I Will” and “Julia”
Two of the simplest, most beautiful songs in the Beatle’s oeuvre, IMHO.
Living Prayer by Alison Krauss. (opens in Windows Media Player.)
Listen to Patty Griffin’s album “Living With Ghosts”. Do not pass Go.
Lori McKenna’s “The Kitchen Tapes” is also good, albeit even rawer.
I’d add When You Come Back Down and many, many others. 
You’ve Got to Hide Your Love Away - The Beatles
Yesterday - The Beatles
Leaving on a Jet Plane - written by John Denver, performed best by Peter, Paul, and Mary.
Hearts and Bones - Paul Simon
*In This Life * - Collin Raye
*Pili Me Ka-u Manu * - Iz (aka Israel Kamakawiwo’ole)
and a gratuitous plug for someone I know (kinda):
Harbor - Vienna Teng (Live version at NPR) or (CD version from her site)
Sounds like you might enjoy some folk and/or bluegrass. When I read the title looking for “simple and beautiful” songs, the first thing that popped into my mind were some old Shaker melodies – “How Can I Keep From Singing,” for example.
“I’ll Fly Away” is a beautiful, simple song; there’s a very nice duet version of it on the CD “Down From the Mountain” (kind of an “O Brother Where Art Thou” tie in).
And given the OP’s name, you don’t think “Mother Nature’s Son” is a good choice?
It’s called… ready?.. “We’re Going to Be Friends.” 
Yep, absolutely. Hadn’t thought about that…I was just focusing in on CD 1 (LP 1/Side 2), I guess.
[newbie] Are we allowed to say “lol” in here? [/newbie]
I love that song. I just heard Jack Johnson cover of it, not too bad.
Pretty much the entire catalog of Gordon Lightfoot. Start with “If You Could Read My Mind.”
I’ve got a noggin overstuffed with songs, but the most simple lala beauty one for me is George Harrison’s Here Comes The Sun
It’s Alright…
I was just about to mention her in this post. I love her voice. She is one of the few artists who could make a cover song better than the original.
As for yet another cover recommendation I also love the simplicity of Iz’s “What a Wonderful World/Over The Rainbow”. I know it has been played over and over again, for that damn commercial, but I still love it.
Dirty Vegas, Days go by.
Yeah I know its techno but theres a version out there with just a singer and a guitar thats fantasticly simple.
Chris Cornel, Seasons.
from the Singles sound track, great song that really puts his voice on display.
theres a couple off the top of my head.
Alot of these listed are favorites of mine. I’m going to have to check out some of the ones I haven’t heard!
My contribution is “Rhoda” by Belle & Sebastian. I don’t know that it’s ever been released on an album or not. Don’t think it has.
Alot of their stuff is very simple and beautiful. Then again there’s about 8 people in the band most times so some of their things are very intricate and beautiful.
One of my contributions is Alison Krauss’ “When You Say Nothing At All.” What a voice.
Simple and beautiful? Check out Marvin Gaye’s"Star-Spangled Banner" from the 1983 NBA All-Star Game. Seriously.
“Life By the Drop” - Stevie Ray Vaughan.
“When Doves Cry” - Prince.
“Sweet Jane,” “Misguided Angel” - The Cowboy Junkies.
Celine Dion - “Calling You” (from 1994, back before she got David Fostered).
“The Blower’s Daughter” - Damien Rice
I’m surprised no one’s mentioned Nick Drake, but pretty much anything he ever did qualifies. Pink Moon is a favorite. Others:
Why Worry - Dire Straits
Good Riddance - Green Day
Hallelujah - Jeff Buckley
Fell in Love - Moxy Fruvous
Blizzard of 77 - Nada Surf
Valentine - Old 97s
Question - Old 97s
Take Off Your Cool - Outkast
Quiet - Rachael Yamagata
Skyway - The Replacements
Alright for Now - Tom Petty
In Perpetuity - Too Much Joy
Your Dictionary - XTC
The Only Answer - Mike Doughty
I agree wholeheartedly.
But did you know it’s a cover, too?