They would likely search for him. His mother, in particular, would be heartbroken. She would scour every corner of the village and beyond, hoping to find her son. This could lead to the family members becoming closer as they unite in their search for Naruto. Or it could cause more rifts if they start to blame each other for his departure.
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.
His departure could also change the family's reputation in the village. People might start to gossip about what could have driven Naruto away. This could put pressure on the family, and they might have to face the judgment of the villagers. On the other hand, it could also make them more determined to set things right and bring Naruto back to show that they are a loving and united family.
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.
Some of his friends could be inspired. They might see Naruto's departure as a noble act, especially if he's leaving to protect his family. Naruto has always been a source of inspiration, and this could lead them to re - evaluate their own relationships and responsibilities. For instance, Sakura might be motivated to be stronger to support Naruto's family in his absence.
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.
Naruto might rely on his ninja skills that he's learned. He's a resourceful character, so he'd use his stealth and combat abilities to find food and shelter. For instance, he could hunt small animals in the forest and build a simple shelter using his knowledge of basic survival techniques.
The plot could take a very emotional turn. His family might be heartbroken and try to find him, which can drive a big part of the story. Meanwhile, Naruto himself could be going through a lot of self - discovery. He might become stronger or change in ways that are both positive and negative as a result of leaving his family. This can then have a huge impact on how he interacts with the world and the people he meets later in the fanfic.
One possible plot could be that Naruto feels constantly overshadowed by his family's expectations. Maybe his family is full of powerful shinobi and they expect him to be just as great immediately. He runs away to find his own path and prove himself on his own terms. Along the way, he meets new friends who are also outcasts and together they form a sort of 'family' that accepts him for who he is.
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.
It might be that his family is overprotective. Naruto is a very independent character. If his family in the fanfiction is constantly trying to hold him back from taking risks or going on adventures, he may feel suffocated. Running away would be his way of breaking free and exploring the world on his own terms.
He might meet an old hermit who teaches him a unique and powerful jutsu that his family never knew. This new jutsu gives him an edge and also makes him more confident in his abilities. He could then use this newfound power to help those in need, all while staying away from his family.
His departure would definitely put a strain on his relationships. His close friends like Sasuke, if he is back in Konoha at that time, would have mixed feelings. They might worry about Naruto but also be a bit angry that he left without consulting them. Naruto's younger friends like Boruto would be confused and miss him a great deal, and this could lead to some distance in their relationships.