He might feel angry and frustrated. Naruto has always been proud of his potential and his roots in some way. Being denied his heritage would be like taking away a part of his identity, so he'd likely be quite mad at those who are doing the denying.
One consequence could be a loss of self - confidence. His heritage has been a part of what has motivated him to become stronger. Without that connection, he may start to doubt his abilities.
He might be very disappointed at first. Naruto had always dreamed of becoming Hokage, so it would be a huge blow. But he could also use this as motivation to prove himself even more. He would likely go on some self - improvement journey, training harder and showing the village that they made a mistake.
Naruto could be confused. He may not understand why he was denied when he has accomplished so much. He might start to question the system and those in power. This could lead him to have some internal conflict as he tries to figure out whether he should continue to strive for the promotion within the existing framework or find his own path.
In the fanfiction, when Naruto learns of his heritage, he often becomes more confident. Knowing where he comes from gives him a sense of purpose. He might start to train harder, believing that he has a greater destiny due to his lineage.
Well, it could strengthen his bond with some characters. For example, he might feel a closer connection to his clan members if he discovers his heritage. He could also gain new respect from his peers who now see him in a different light, perhaps as a more important figure within the village due to his lineage.
Naruto might initially be confused. He has grown up with a certain view of himself, and this new information about his heritage completely shatters that. He may struggle with his identity for a while, but as he digests the information, he could start to feel a sense of pride. He might think that now he has a greater purpose, something that ties him more closely to his family and the ninja world in general. He may also become more confident in his abilities, knowing that he comes from a great lineage.
Well, it could strengthen his bonds. His friends might see him in a new light, with more understanding and respect. For example, they could realize the weight of his past and be more empathetic towards his struggles.
It would also impact his relationship with his village. The village has always seen Naruto as a part of their heritage in a way, especially considering his connection to the Fourth Hokage. If he rejects his heritage, some villagers might feel betrayed or confused.
Naruto might initially be very disappointed. He has always been eager to train and become stronger. So being denied by Kakashi could make him feel a bit down. But he could also be defiant. He might start training on his own, determined to show Kakashi that he doesn't need his approval to get stronger.
He might start seeing himself as part of a greater legacy. Knowing he's an Uzumaki could make him feel more powerful and connected to his roots. He could see that his strength isn't just his own, but part of a family trait.