I guess it depends on your body fat % but your BMI is only 28.4 so I doubt it's a weight problem that caused your back injury. I'm pretty skinny and I know my BMI is over 27, if any doctor called me fat I'd give him a slap (joke)
How long has your back been hurting?
Does the pain run down one of your legs?
Is the pain constant or just when you move in certain directions?
I've had a history of back problems, my doctor was actually pretty good (although I didn't think so at the time), he told me that I was just lazy and all the pain was being caused by over stressed muscles.
My girlfriend wasn't so fortunate, she had a herniated disk, this is where the disk pops out briefly and pops back in again, it causes damage and needs a lot of rest followed by light exercise to build up surrounding muscles.
If the disk stayed out of place then it's a slipped disk and you would be having trouble even walking. If the pain is bad then get a second opinion asap as the long this goes on untreated the worse it'll get