After Jerry successfully swims through the tunnel, he feels he has accomplished what he set out to accomplish. He has proven his ability - he has demonstrated his "manhood," so the bay is no longer important. He doesn't need to prove it again.
Getting to this point, though, took a lot out of him. All the hours in the water, the practice holding his breath, the nose bleeds, the head aches, the lies to his mother, and finally the fearful trip through the tunnel - it all leads to that moment when he feels satisfied. The earlier laugher of the older boys is drowned out, and instead he hears his own voice saying, "Well done."
When, exhausted, he finally reaches the shore after swimming through the tunnel, he takes a moment to catch his breath and rest:
"He scooped up handfuls of water from the cool, salty sea, to splash on his face, and did not know whether it was blood or salt water he tasted. After a time, his heart quieted, his eyes cleared, and he sat up. He could see the local boys diving and playing half a mile away. He did not want them. He wanted nothing but to get back home and lie down."
After this, he returns to the villa to rest, sleep, and clean himself up. When his mother sees him and sees how "strained" his face is and how "glazed-looking" his eyes are, she worries and warns him not to overdo it. As the author writes, "She (is) ready for a battle of wills, but he (gives) in at once. It was no longer of the least importance to go to the bay."
Jerry has proven what he needed to prove - not to the other boys, but to himself. He has proven his maturity - he has grown up and passed "through the tunnel" into manhood - and he can now relax with his mother, feeling satisfied with his accomplishment.