Well, no. You can't really prove the theory of music. The proof of the pudding is in the eating.
Pythagoras probably wasn't the first human to speculate on the relationships of pleasing sounds, but he may well have been the first to appreciate the value of integer ratios in string length (or he may just be the earliest case we have records about).[*]
As to your second question, he may well have been "the first one who invented a musical scale". As, by formalizing what was formerly done intuitively, he may have been the first to accurately (reproducibly) define a scale. But other scales already existed, having evolved with the culture of music, but possibly never having been invented.
Another complication is that as the founder of a school, many works of other members of the school were attributed to the master. As Plato puts all his ideas in the mouth of Socrates.
[*] Many of the stories describe Pythagoras deriving the formula after hearing hammers striking different-sized anvils or pots, but these do not follow the same law (as the pitch of a struck anvil or pot depends upon mass or volume, not simple length as the formula has come down to us). Some event like this may have inspired him to investigate lengths on the monochord, but he almost certainly didn't come up with the formula merely by walking past a blacksmith shop.