Well, it makes him more independent. Since he's left behind, he has to learn to fend for himself. He becomes more self - reliant and determined to prove his worth, not just to his family but to the whole village. He's constantly seeking validation and that drives him to train harder and grow stronger.
Well, Naruto could become overly self - reliant. Since his family left, he may think he can only count on himself. So he might push people away who try to help him. This over - self - reliance could also make him stubborn in his ways and reluctant to accept advice. In fanfiction, this could be explored in his training. For example, he might refuse to learn new techniques in the way his teachers suggest because he believes he can figure it out on his own due to his abandonment issues.
Iruka could step in. He has always been like a mentor and a father figure to Naruto. He was the first one to really see Naruto's potential and give him the recognition he deserved. So in the absence of his family, Iruka would be a likely candidate to fill that family - like role.
One possible reason could be that his newfound 'godlike' powers make his family feel estranged. They might not understand his sudden transformation and the magnitude of his power, which creates a sense of fear or discomfort, leading them to ignore him.
Naruto could develop a stronger sense of justice. Because he was left by his family, he may have a greater desire to protect others from experiencing the same kind of abandonment. He would strive to create a world where families stay together and no one is left alone. This could be shown through his actions in the fanfiction, like him going out of his way to help orphans or families in distress.
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.
He would probably be very lonely. Without a family, Naruto has no one to turn to in the most basic sense, which would lead to a deep - seated loneliness.
Maybe he has some personal grudges or misunderstandings within the family in the fanfiction. It could be that something happened in the past that made him feel betrayed or unappreciated, so he refuses to offer help.
Naruto in such a fanfiction could have a very different growth trajectory. Abandoned by family, he might be more eager to find a sense of belonging elsewhere. He could be more drawn to groups or individuals who seem to accept him unconditionally. His view of the village might also be more cynical. He may see the village as a place that failed him by allowing his family to abandon him. His training could be more solitary as he doesn't have a family to guide or encourage him. This would likely lead to a Naruto with a more independent and perhaps more hardened personality compared to the Naruto in the original story.
Perhaps in this fanfiction, Naruto's family leaves him because they fear his godlike abilities. Naruto then has to prove that he can use his powers for good. He could start by helping the villagers who once shunned him. This would lead to a series of events where he gradually gains the respect and understanding of those around him, despite his family's absence.