In some cases, Naruto might initially become bitter towards his parents. But as he experiences more in life and learns about the possible reasons behind their actions, he could come to understand and even forgive them. His reaction would also depend on the people around him. If he has good friends or mentors, they could help him deal with his emotions in a more positive way.
Minato might be intrigued. He would be interested in seeing how the fans expand on Naruto's journey. Kushina would be delighted, seeing her son as the center of so many different tales.
At first, Naruto and Kushina might start to distance themselves from Minato. They could avoid him and not talk to him as much. As time goes on, the hatred might grow deeper. Naruto could start to oppose Minato in public or in important village matters. Kushina might also try to influence others in the village against Minato.
The story could also show Naruto being taken in by other characters, like Iruka at a very early age. Iruka, seeing Naruto's situation, decides to be like a father figure to him. Naruto might then develop a different set of values based on Iruka's teachings rather than the absent influence of Minato and Kushina. He could become more focused on family values within the village, seeing the village as his true family since his biological family left him.
Naruto might initially be very confused and hurt. He loves his father and the village, so being banished would be a huge shock to him.
Perhaps it has to do with a prophecy. They might think that Naruto has a special destiny and that their leaving is a necessary step to set that destiny in motion. They could believe that Naruto needs to grow up on his own to become strong enough to face future challenges.
If Minato and Kushina abandon Naruto in a fanfic, it could lead to a much darker Naruto. He might grow up more jaded and bitter. Without their love and support, he could be more easily influenced by negative forces. Maybe he'd have a harder time believing in friendship and loyalty, which are such important themes in the original story.
Naruto could be found by Orochimaru. This would be a very dark turn. Orochimaru could use Naruto's potential for his own evil plans. Naruto would be trained in Orochimaru's unethical ways, and his view of the ninja world would be completely distorted. He might become a villain or at least an anti - hero, very different from the Naruto we know in the original story.
Naruto might be extremely heartbroken. He has always looked up to his mother, and being disowned by her would be a huge blow. He could become withdrawn and start to question his own worth.
Kushina would be extremely worried. She would rush to Naruto's side immediately and start using her medical knowledge or any available resources to help him get better.
As he grows up, Naruto could become very angry. He might develop a sense of abandonment and this could fuel his determination to become stronger. He would want to prove himself to his mother, wherever she is. In this fanfiction, his anger could be a driving force in his character development.