It could also be due to some external factors. For example, the village elders might put pressure on Naruto to stop trying to save Sasuke because they see Sasuke as a threat. Naruto might be forced to choose between the village's safety and his friendship with Sasuke, and in the end, reluctantly give up on Sasuke.
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.
One possible reason could be that Naruto realizes Sasuke is too far gone in his quest for revenge. Naruto might see that no matter how much he tries, Sasuke won't change his ways, so he gives up. For example, if Sasuke keeps attacking innocent people in his pursuit of power to defeat Itachi's enemies, Naruto might finally lose hope.
In some fanfiction, Naruto could give up on Sasuke if he falls in love and his new relationship makes him see things differently. His new partner might convince him that Sasuke is a lost cause or that he should focus on their own future instead. Also, if Naruto suffers a major setback in his own life, like losing a dear friend in a fight related to Sasuke, he could lose the will to keep chasing Sasuke. This would be a very emotional and complex turn of events in the fanfiction world.
It could also be due to a misunderstanding. Some false information might have spread in the fanfic's world, making Naruto believe that everyone has turned their backs on him. Like, some bad guys framed his friends, and Naruto, being unable to see through the lies at first, gives up on everyone in despair.
One possible reason could be that Naruto realizes Sakura's true feelings for Sasuke are unwavering. After many attempts to win her heart, he might come to the conclusion that he's just chasing after someone who will never love him back. So, he decides to give up and focus on other aspects of his life, like his ninja duties or his friendships with other people in the village.
One possible reason could be that Naruto realizes Sasuke is too far gone on his dark path. After numerous attempts to bring him back, Naruto might start to think that Sasuke doesn't want to be saved. He may see that Sasuke's actions are causing too much harm to the village and the people he loves, and Naruto has to prioritize them over Sasuke. So, he gives up on the idea of bringing Sasuke back.
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.
Well, perhaps it's because of Sasuke's aloof nature. Naruto is a very open and friendly person, and constantly feeling shut out by Sasuke could lead to a break - up in the fanfiction. Also, Sasuke's thirst for power might be too much for Naruto to handle. Naruto believes in the power of friendship and hard work, but Sasuke seems to be constantly on a solo mission for more strength, which could create a rift between them.
If Naruto gives up on Sasuke in fanfiction, the storyline could take a very different turn. For one, the whole dynamic of their relationship, which has been a central part of the original Naruto story, would be completely disrupted. Naruto has always been determined to bring Sasuke back to the village. Without that drive from Naruto, Sasuke might become even more isolated and perhaps fall deeper into the darkness. This could also lead to different power alignments in the ninja world. Allies of Naruto might question his change of heart, and enemies could take advantage of the rift between the two former friends.
One possible reason could be that Sakura realizes Sasuke is too consumed by his revenge and she can't keep waiting for him to change. She might see that he is constantly putting her in a secondary position and she decides she deserves better. For example, Sasuke has left the village multiple times, leaving Sakura heartbroken each time.
The dynamic between Naruto and Sasuke would be completely different. Instead of the usual pursuit, there would be a sense of finality. Naruto might start to avoid Sasuke, which would change the way Sasuke views Naruto too.