On the positive side, his anger could also bring him closer to some people. There might be others in the village who also had difficult relationships with their parents or family. Naruto's situation could make them bond over shared experiences. However, overall, his anger towards his parents is likely to cause some upheaval in his relationships until he can come to terms with it.
It might make him more distant from his friends at first. He could be so consumed by his anger towards his parents that he withdraws from his usual social circle. For example, he might avoid spending time with Sasuke or Sakura because he is too preoccupied with his feelings about his parents.
Perhaps Naruto has always felt different from others in the village. And when he finally learns about his parents, he realizes that their actions, even if they were for a good cause like protecting the village, still made his life very difficult. He might be angry that they didn't find a better way to handle things so that he didn't have to suffer so much growing up.
His relationships with his peers can become more intense. For example, with Iruka - sensei, Naruto might cling to him more as a father figure. With his teammates, he may be overly eager to prove himself worthy of their friendship. He could be more sensitive to any signs of rejection from them.
It could make him very distrustful at first. He might have a hard time believing that others truly care about him. For instance, when his friends try to be nice, he might think they have ulterior motives.
It might make his relationship with his friends a bit strained. For example, Hermione and Ron might not understand his anger at first and try to convince him that his parents were heroes. This could lead to some arguments between them.
It would likely make his relationship with Iruka even more important. Iruka was one of the first to show Naruto kindness, and with his parents' absence, he would cling to Iruka as a father figure even more. Naruto might also be more eager to prove himself to his peers. He would want to show that he doesn't need his parents and can be just as good or better than those with a normal family. This could lead to some over - competitiveness at times.
It could strain his relationship with Sirius. Sirius was his father's best friend, and if Harry doesn't forgive his parents, Sirius might feel caught in the middle. He might try to defend them, which could lead to arguments between him and Harry. For example, Sirius might talk about how brave his father was, but Harry, in his state of non - forgiveness, might not want to hear it.
It could be that in a special mission, Naruto discovers a hidden jutsu that allows him to communicate with the spirits of his deceased parents, leading to a touching reunion.
He could also distance himself from them. Naruto might choose to avoid his family, not wanting to be around them as a way to show his anger. He could move out or stop participating in family activities.
On the positive side, it could bring him closer to those who also have issues with his father or the old ways. There might be some characters who have been wronged by Naruto's father's decisions in the past. Naruto's hatred could make them form an alliance of sorts, united by their shared negative feelings. However, overall, it would likely cause more problems than benefits in his relationships.
His relationships might be less intense in some ways. Since he has a family, he might not be as clingy to his friends like Sasuke. For example, he wouldn't be as desperate to bring Sasuke back because he has his parents' support at home.