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.
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.
In some fanfictions, Percy could be made Artemis' son for creative and often unique plot twists. Fanfiction allows for a lot of creative liberties, so it's not unheard of. However, in the original Percy Jackson lore, there's no such relation.
It would be complex. Artemis, as his mother, would likely be protective yet conflicted. She has her own identity as a virgin goddess, and Percy's existence challenges that. But she would also love him deeply. Percy would probably look up to her and be in awe of her powers, yet also struggle with the fact that his mother is such an unusual figure in the mythological world. There might be a sense of distance at times because of Artemis's nature, but also an unbreakable bond.
Their relationship might develop through facing external threats. For instance, if there are enemies that target them because of their forced union. They would have to rely on each other's skills. Percy's ability to control water and his combat skills, along with Artemis's archery and connection to the wild, could be used to fend off these threats. As they fight side by side, they start to see each other in a new light and develop a bond.