Hmm. using Saitou’s earlier link I found the Marvin the Martian bio has a suspect:
Spaced-Out Bunny[scroll down]–
Yeesh beat by 5 min? Shoulda previewed and bailed…
Ok…I’m done googling. (Really I am.) But it seems you weren’t the only one curious Qadgop the Mercotan 
Daizy
It was used in the 1946 Tex Avery cartoon Lonesome Lenny. Lenny, the dog, wants a friend and chooses Screwy Squirrel. Not surprisingly, it’s the last Screwy Squirrel cartoon.
Sad ending. Wasn’t it?
But he don’t move no more.
Okay, I also remember hearing that line in a Bugs Bunny cartoon with Wiley Coyote and the really hairy sheepdog. The sheepdog was guarding a herd of sheep (naturally) and Wiley Coyote was trying different ways to nab one of the sheep.
I think (and I am probably dis-remembering it completely) that the sheepdog got bonked on the head with something and was a little woozy and grabbed a sheep, started petting it and saying, “and I will love him and squeeze him and call him George!”
Or maybe that was Wiley Coyote.
Or maybe I just made it all up with my vivid imagination at the time.
I DO remember in that same episode that in the beginning Wiley Coyote and the sheepdog punched in on a timeclock and told each other good morning before chasing each other around all day. That always cracked me up. 
I remember it from the cartoon with the Abominable Snowman – specifically, the one where Bugs and Daffy are trying to get to Pismo Beach and wind up in the Himalayas.
My brother and sister and I named our cat ‘George’ after someone remembered this cartoon
Greywolfe the Abominable Snowman was drawn almost identically to the sheepdog. Only he had white bangs instead of red bangs in fron of his eyes.
(Favourite moment: when Bug tricks Daffy into wearing a rabbit suit, and Abominable spanks the “naughty, naughty bunny rabbit”.)
Somehow (I’m probably mis-remembering) I remember this “dumb dog” line being in the cartoon where Bugs Bunny is trying to evade a pack of fox hounds and the big dumb dog is the laggard in the pack. At some point he finds Bugs and thinks he is fox and utters this line…or not.
I remember the snowman one. That’s where I got it when I decided to name my now sadly deceased leopard gecko George. He was a cute lil’ lizard guy!
Mindy and Buttons?
Nitpick: That’s not Wile E. Coyote in the cartoon(s) with the sheepdog. The noses are different colors. That’s a wolf, and they say his name in the cartoon (though I don’t remember it).
The wolf’s name is Ralph (the wolf has a red nose, Wile E.'s is black).
another “Of Mice and Men” Looney tune:
“Dough Ray Me-Ow” (1948, Arthur Davis) featuring Louie the Parrot and Heathcliff the Cat (no “George”, though).
I remember the Bugs/Daffy episode, mostly because I loved that line. And then on the Australian Big Brother 2, one of the first housemates to get voted out was given a goldfish, and, when asked what its name would be, said ‘George. Because I will hug him and love him and squeeze him…’
My brother and I cheered, and felt bad about being glad she was voted out.
The one I remember the most was with the Abominb-b-b-ble Snowman and Sylvester. Especially the part when Sylvester says, “but my name is Sylvester” and the snowman says, “but I can’t say Sylvester, George.”
It’s funny this came up because my youngest is named George–after his dad. When he asks why I named him George I always tell him it’s because when I first saw him I thought “I’m gonna love him, and squeeze him and name him George.” So he’s rather proud of the fact he’s named for a cartoon