Perhaps there's a powerful artifact that can only be activated by a father - daughter bond between Percy and Artemis. They are forced to work together to find and use this artifact to stop a great catastrophe. Along the way, they face many challenges. Percy has to deal with the fact that Artemis is a very different kind of 'daughter' compared to what he might expect, and Artemis has to come to terms with having a father figure for the first time in her existence.
Well, in the original Greek mythology, Artemis is a virgin goddess and having Percy Jackson as her father is a completely new and wild concept. But in fanfiction, anything can happen. It could be a story where some powerful magic or strange twist of fate made it so. Maybe Percy's heroics led to some divine interference that resulted in this unexpected relation.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.
Well, Artemis is a complex character herself. Perhaps she sees something of herself in Percy, like his determination or his sense of loyalty. Fanfiction often takes these qualities and twists them in new and interesting ways. She might be drawn to how Percy is portrayed in different fan - written stories, whether it's his heroic deeds or his more vulnerable moments. Reading Percy fanfiction could be a form of escapism for her, a way to explore a different kind of heroism outside of her own experiences.
One reason could be the popularity of Percy as a character. Fans might be more inclined to write about him in a dominant role because they like him better. So, they create fanfictions where he dominates Artemis.
Perhaps a story where Percy is chosen by the Fates to mediate between Hestia and Artemis. There has been some sort of misunderstanding between the two goddesses. Percy, with his charm and unique perspective, tries to get them to see eye to eye. He arranges meetings in different places - sometimes in Hestia's warm and cozy domain, and other times in Artemis's wild and untamed forests. Through Percy's efforts, they not only resolve their differences but also realize that they can work together for the greater good.