You are right, in modern times, carbs are generally considered bad. But some people trying all protein diets do report diarrhea. Doctors don't generally refer to it as too much protein, though. They refer to the problem as too little fiber.

Also, some arctic explorers tried living on all meat, and suffered vitamin deficiencies such as scurvy. So, while meat is a complete protein, unless you are eating fresh liver meat from a fresh kill, you are going to need some source of vitamin C at the minimum. Except for liver, meat isn't actually a great way to get fat soluble vitamins either.