He might be angry at first. Naruto has grown up alone and had to fight for acceptance. So when his parents say they regret abandoning him, he could lash out, saying it's too late. After all, he has already faced so much without them.
The son could ask for some time and space if he feels too overwhelmed in the moment. Then, when he's ready, he can sit down with his parents and have a proper conversation. He can also share some educational resources about homosexuality with his parents to help them understand better.
In some fanfictions, Naruto may become extremely bitter. He could feel a deep sense of betrayal and this might fuel his determination to become stronger, not for the sake of the village at first, but more for self - vindication. He might isolate himself from others and struggle with his identity.
In some versions of the fanfic, Harry might have a sort of instinctive understanding that his parents had a good reason for leaving him there. So, while he's still not happy about it, he might be a bit more accepting. Maybe he can sense some sort of protection spell around the Dursleys' house that gives him a small sense of security.
Another fictional hawk could be 'Hawkeye' which is a name that can be associated with a hawk - like character. It gives the impression of a character with sharp vision and a focused, predatory nature, much like a real hawk.
Some adults may feel extremely humiliated. They are no longer children and being spanked makes them feel as if their parents don't respect their adulthood. For example, they might become withdrawn and avoid family interactions for a while.
One possible reason could be that Naruto has some sort of power within him that they fear. Maybe it's a power that they don't understand and think could be dangerous, so they hate him out of fear.
I recommend "The Peasant Daughter-in-law". The female protagonist of this book also raised a child alone, while the male protagonist was the eldest young master of a peasant family. The plot also involved the situation where the female lead avoided certain people in order to protect her child. The entire novel was set in business and farming, telling the story of how the female protagonist started her own career and found true love. I hope you like my recommendation, Mwah~
The Hawks' back story could involve their origin as a sports team. They might have started in a particular city with a group of passionate players and investors. Maybe they were initially part of a smaller league and then grew to be in a major one. Their name could be inspired by the characteristics associated with hawks, like speed and precision.