Jerry's having swum through the tunnel is his rite of passage into manhood, and that is enough.
By overcoming great obstacles and by facing danger alone, Jerry has acquired greater maturity and independence. In fact, telling his mother what he has accomplished could mitigate, or reduce, Jerry's personal pride in his newly acquired manhood. For one thing, she might display anxiety about his having tried such a dangerous feat, and then Jerry could feel some guilt about his actions. So, when he and his mother sit down to have lunch together right after she has "looked at him closely" and has seen that he is "strained," Jerry probably perceives the worry and anxiety in his mother's face. Feeling the need to relieve her anxiety, as well as having a child's need to tell his parent about an accomplishment, Jerry does reveal to his mother that he can now hold his breath for at least two or three minutes.
This revelation brings from his mother both praise and concern:
"Can you, darling?" she said. "Well, I shouldn't overdo it. I don't think you ought to swim any more today."
Jerry perceives from this response that he has made the right decision in choosing not to inform his mother of his feat. He wants his rite of passage to remain free of emotional and parental entanglements so that he can maintain his newfound independence.
In the end, Jerry doesn't tell his mother about swimming through the tunnel because what he seems to want, at that point, is her approval and praise, and he likely knows that she would not approve of his decision to put his life in danger in order to do something so completely unnecessary and stupid.
In the beginning, Jerry understands that his mother worries about him. "He was very familiar with that anxious, apologetic smile" of hers that signaled her twin worries: worrying about giving him too much freedom and worrying that she doesn't give him enough. Therefore, in the end, to tell her that he had done something so dangerous would absolutely confirm that she has erred by allowing him too much independence. He does not want to make her worry because he feels somewhat "chivalr[ous]" towards her and experiences the pangs of "contrition" when he thinks he's letting her down.
Further, he does want her approval and praise, and so he impulsively tells her that he can "'stay under water for two minutes -- three minutes, at least.'" The narrator says that "It came bursting out of him" as some such news would from a little child. This is an achievement that he feels safe telling her because it wouldn't endanger him or cause her to worry. Like a child, he longs for her praise, and this is an achievement that she can (and does) praise.