They would probably start searching for him. This could lead to the family members getting closer as they united in their goal to find Naruto. Or it could cause conflicts among them if they blamed each other for Naruto's running away.
In the long term, if Naruto doesn't return, the family might have to reevaluate their values and the way they treated Naruto. They might change their ways in hopes that if Naruto ever comes back, they can have a better relationship. Also, Naruto's friends might get involved and try to bridge the gap between Naruto and his family.
Some of his friends might feel betrayed. They could think that Naruto didn't trust them enough to share his problems about his family. For example, Sasuke, who has his own issues with family, might initially be angry at Naruto for running away instead of facing the family situation together like they face other challenges.
The family would be shocked at first. They might start to question their actions towards Naruto. If they were too strict or didn't give him enough love, they would feel guilty. For example, his father may realize that he pushed Naruto too hard to follow a certain path and now regrets it.
Running away from his family can make Naruto more independent. He has to rely on himself, which would strengthen his self - reliance. He'll learn to make decisions without the influence of his family, and this could make him a more decisive ninja.
It could be due to a family secret. Suppose his family was involved in something dangerous or unethical that Naruto couldn't bear. For example, if they had ties to an enemy organization that Naruto discovered. He is a character with a strong sense of justice, so running away would be a way to distance himself from their actions and figure out how to deal with the situation on his own.
One possible reason could be that he felt misunderstood. In the Naruto series, Naruto was often an outcast in the village. Maybe in this fanfiction, his family didn't understand his dreams or his need for recognition. So he ran away to find a place where he could be himself.
The villagers who once shunned Naruto might start to realize his importance. They would see how much they took his presence for granted. His absence could make them reflect on their past behavior towards him. Some might even join the search parties out of guilt or new - found respect.
It would likely create a lot of tension. Family members might be confused and hurt by his coldness. They may try to reach out to him but be constantly rebuffed, which could lead to feelings of sadness and helplessness among them.
However, it could also have negative impacts. He might start to feel lonely and bitter, which could lead to him making some bad decisions. He could be more easily influenced by the wrong people or ideas because he lacks the grounding that his family usually provides. Also, his relationships with his friends might become strained as he tries to keep them at a distance too, thinking that they'll side with his family.
It would create a sense of panic. Naruto is a very important figure in Konoha. His strength has been crucial in protecting the village. Without him, the villagers would fear attacks from other villages or threats from the Akatsuki. For instance, if the Akatsuki knew Naruto had left Konoha, they would likely see it as an opportunity to capture him more easily.
From a security perspective, Konoha would be weaker. Naruto is a very powerful ninja, and Hinata is no slouch either. Their skills and abilities would be missed. Enemies might see this as an opportunity to attack Konoha. Also, Naruto has been a symbol of hope for the village. His departure could lead to a decrease in morale among the villagers. They might start to feel less secure and more pessimistic about the future.