One possible reason could be that he experiences a great loss, like the death of a dear friend. This might shatter his confidence and make him lose his passion for being a ninja.
Another factor could be the nature of his ninja life. It's full of uncertainties and risks. He may worry that if he falls in love, his loved one could become a target for his enemies. Naruto has made many enemies over the years, and he wouldn't want to put someone he loves in danger, so he decides to give up on love.
One reason could be the constant setbacks he faces. He might feel that no matter how hard he tries, he can't achieve his goal. For example, if he fails too many times in battles against strong opponents, he could lose hope. Another reason might be the loss of important people in his life. If someone like Iruka or Kakashi, who have been his pillars of support, were to die, it could shatter his dreams as they are so closely tied to his will to succeed for their sake.
He becomes more solitary. Without the pursuit of love, Naruto focuses more on self - improvement and his ninja skills. He might seem more aloof as he distances himself from the idea of a relationship. For instance, he spends more time training alone in the forest.
The story would take a very different turn. The dynamic between Naruto and the rest of the characters would change. For example, Naruto might become more focused on protecting the village without the distraction of Sasuke. His relationships with other friends like Sakura might also be affected. Sakura might be disappointed in Naruto at first for giving up, but then they could bond over the loss of Sasuke in a different way.
If Naruto gives up on his dream in a fanfiction, the whole Naruto universe would be drastically different. His friends would be shocked. For example, Sasuke might not have the same motivation to grow stronger as Naruto's unwavering determination was always a driving force for him. The village of Konoha would also lose a great source of inspiration. Naruto was always the one to believe in the village's future and strive for its betterment. Without his dream, the hope he spread might fade away, leading to a more gloomy atmosphere in the village.
The story could develop in a way that Naruto's friends try to cheer him up and convince him to re - enter the exam. They might show him how much they believe in him and that his giving up is not the solution. For example, Sasuke could tell Naruto that he needs him as a rival to become stronger. Another development could be that Naruto starts to train secretly and then surprises everyone by coming back stronger later in the exam.
One reason could be the sacrifices he has to make for the village. Naruto is so dedicated to protecting Konoha that he might feel he doesn't have time for love. His responsibilities as Hokage are immense, and he has to constantly be on guard against threats. Another reason could be past traumas. He has lost many people he cared about, like Jiraiya. This could make him afraid of getting close to someone again and losing them, so he gives up on love.
Maybe Naruto gets tired of always chasing after Sasuke. He has his own life and dreams too. He might think that he has sacrificed enough of his time and energy on Sasuke, and it's time to focus on himself and the village. He could start to believe that Sasuke has made his own choices and he can't keep forcing Sasuke to come back.