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.