Does rock climbing build chest?

The chest muscles are some of the only muscles that are not directly worked when climbing. Climbing does build many muscles in the body, even legs to some extent, but the chest muscles are just barely used while climbing.

While climbing, your body is under general tension, and there are specific moves that do use the chest muscles, but they don't occur so often as to actually give your chest a good enough workout.

This is why climbers should also include antagonist training and exercises into their schedule to help build the muscles that are not really worked enough through climbing, otherwise they could suffer muscle imbalances that eventually could lead to bad posture and to injury.

Not really. Climbing is more of a pulling exercise for the upper body. This basically leaves your chest and triceps sort of neglected, which is why many climbers do antagonist training such as push-ups, dips, and bench presses.

There are some climbing moves such as mantles that use your triceps, and to some very slight extent, your chest. But for the most part, your chest is not used enough to build it a lot, at least not as much as your other muscles are used, such as your back. Which, again, will cause an imbalance in your muscles if you don't train your chest and triceps specifically. Your posture could be effected by this quite a lot.