Rabbits are, by nature, herbivores...not vegetarians. I give mine pellets, hay and then grass or greens/weeds or leaves from plants that they can eat. I don't feed them root vegetables like carrots, but I have fed the carrot tops. I don't give them much fruit, unless it is something like the strawberry tops or rinds from watermelon.
I feed carrots, celery, or lettuce in small amounts every evening. My rabbits love it and have had no problems with digestion. Garden time is coming and i hope to plant several things for the rabbits such as radishes for the tops. Lots of weeds here so will do a little research to see which they like and can tolerate.
There are a lot of mixed signals out there. I am relatively new to this venture, but I feed pellets everyday, plenty of hay, but at night I give a handful of greens (endive, kale, comfrey, etc.) For my 'feeder' rabbits (young ones raised for meat) I keep on the grass in my yard, feed them hand fulls of everything I get get out of the garden that is weedlike, or greens, which I grow in plenty. For breeders, I just keep it to a reasonable amount, only at night and cut back if i see any diarrhea or if they are not eating pellets. All that said: i would love to go 100% natural, but for breeders i think it is too risky until I figure it out better.
It is perfectly fine to feed your bunny fruits and vegetables everyday. I feed my bunny, Flop, fruits and vegetables everyday and he's happy and healthy. 4 cup vegetables and 2oz of fruit, but if you miss one day of not feeding him this, okay.