Interesting article, but I think you got mixed up in the beginning by saying:
He noticed that the water displaced by his body was equal to the weight of his body.
What I think you meant is that the volume of water displaced by an immersed body is equal to the volume of the latter. Therefore, Gold being twice as dense as silver, he could easily verify if the crown was indeed pure gold - by determining the volume of the crown and knowing its weight.
Thanks for the correction.