Jerry does change significantly from the beginning to the end of the story, but I would not say that he ends up "a confident young man" just yet. Initially, he is anxious to fit in with the big boys, and he immaturely clowns around in order to keep their attention....
It doesn't work. He realizes that he needs goggles to swim through the hole in the rock, and his inability to delay gratification is a further proof of his lack of maturity. His mother agrees to his demand, but he is still impatient:
But now, now, now! He must have them this minute, and no other time. He nagged and pestered until she went with him to a shop. As soon as she had bought the goggles, he grabbed them from her hand as if she were going to claim them for herself, and was off running down the steep path to the bay.
This behavior is extremely childish. However, soon, though Jerry thinks he probably could make it through the rock, he hesitates:
A curious, most unchildlike persistence, a controlled impatience, made him wait.
He is beginning to be able to plan, to think ahead, to exercise caution, and he is more capable of delaying the gratification of his wishes. In the end, he no longer feels the need for the older boys' attention or acknowledgement, but he does still want his mother's approbation and praise. He "burst[s]" to tell her that he can now "stay under water for two minutes—three minutes, at least." This kind of bragging and desire for her notice and praise is still somewhat childish, but the fact that Jerry does not engage her in the "battle of wills" for which she is prepared shows that he has, in fact, matured to some degree.
Therefore, I wouldn't call him a fully mature and confident young man, but he has likely learned a great deal from this experience, which will further his development.