Does rock climbing give you broad shoulders?

Rock climbing can actually be pretty heavy on the shoulders, as they are used in almost all climbing moves, so it makes sense that your shoulders will be strengthened from climbing.

Whether climbing will give you broad shoulders though is another thing. While it's true that climbing will strengthen your shoulders just like any other muscle strengthens due to training, the amount at which your shoulders will grow bigger doesn't depend just on climbing, but also on your body in general and if you also do other exercises and training.

It's very common to see skinny climbers who don't have much muscle mass climbing some of the harder, double-digit routes (v10+ for example), but, it's also common to see bulkier climbers also, some would say that their size comes only from climbing, some will say that it's a combination of climbing and weight training or other combinations.
So some very experienced climbers may not have that much muscle mass, while others may.

In the end, it depends on you and your body. If your goal is to get broader shoulders, then you could try adding weight training to your schedule, if you do not want broader shoulders, you can try just climbing and seeing what that does for you.
It's really very individual and requires experimentation to figure it out.