Yes, he might. Kakashi is a complex character with diverse interests. Given his love for reading in general, it's not a stretch to think he could pick up a romance novel from time to time. He seems like the type who would be open to different genres to expand his knowledge and understanding of various emotions and relationships.
I think his favorite genre is probably about ninja history or combat strategies. Since he is a highly skilled ninja, these types of books would be very useful for him to improve his skills and knowledge. For example, he could learn about the old ninja clans' fighting styles and how to counter them.
Maybe he thinks fanfic often distorts the real story or characters of Naruto. In the original Naruto series, Kakashi has his own understanding and relationship with Naruto. Fanfic might create some scenarios that go against his view of Naruto or the world they live in.
Kakashi would probably be very calm at first. He's an experienced ninja who has seen a lot. He'd gently pat Naruto on the back, offering silent comfort. Then he might start to speak softly, asking Naruto what's wrong in a gentle voice.
At first, it might be a bit rocky. Kakashi is used to his own ways, and fem Naruto may be a bit headstrong. But as time goes on, they start to understand each other. Kakashi sees her potential and determination, and fem Naruto starts to respect Kakashi's wisdom.