Yes, too much fruit juice can cause diarrhea. That's because many juices contain sorbitol, a nondigestible form of sugar. Excess sorbitol levels cause the body to try to dilute the sugar by pulling water from the bloodstream into the intestine, which causes loose stools. That's how prune juice, which is high in sorbitol, helps prevent constipation. Apple, pear, peach, and cherry juice are also fairly high in sorbitol.
The American Academy of Pediatrics recommends limiting the amount of juice your child drinks. Besides causing stomach upset, too much juice can ruin kids' appetite for healthier foods and cause tooth decay. Read more about the pros and cons of serving juice to your child.
Your child needs two servings of fruit per day. If one of those servings is a glass of juice, make sure the other serving is a piece of fresh fruit, which will provide fiber and additional nutrients. Diluting the juice is a good way to make sure your child doesn't drink as much, but between meals (when most "juice abuse" occurs), offer water instead.