animation - Was Bugs Bunny impersonating someone when he ate the carrot?

I've heard that when Bugs Bunny was first shown eating a carrot he was impersonating a popular movie of the time. People of that time got the "joke" and he just kept eating carrots.

In time that movie has been forgotten and now people associate carrot eating with Bugs Bunny.

And the moral of the story is: rabbits don't eat/like carrots normally but they are given them because Bugs Bunny likes them.

Is it true that he was impersonating someone? If so, what was the movie called?