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.
In some fanfictions, Artemis may hurt Percy because of her strict sense of justice. Say Percy breaks one of her rules unknowingly, like entering a sacred forest she has marked off. Artemis, being the goddess of the hunt and protector of the wild, might feel the need to punish him. This could lead to Percy getting physically injured, like being struck by one of her arrows. Or maybe Artemis is under a curse that makes her act out of character and she ends up hurting Percy without really meaning to. This then sets off a chain of events in the fanfiction where Percy has to deal with the injury and also try to find a way to break the curse on Artemis.
One possible scenario is when Artemis criticizes Percy's fighting style. Percy has his own way of battling monsters, which is effective but might not be as refined as the way the hunters of Artemis do it. If Artemis were to publicly criticize his style in front of others, it would surely wound Percy's feelings. He might feel like he is not good enough in her eyes.
For Artemis, show her remorse. After she hurts Apollo, she could be filled with guilt. She might isolate herself from the other gods as she tries to come to terms with what she has done. She could also be shown as determined to make amends, perhaps by seeking out rare herbs or magical items that can heal Apollo. For Apollo, show his pain, both physical and emotional. He might be confused as to why Artemis would hurt him, but also be quick to forgive. He could use his musical and healing powers to try to soothe himself while waiting for Artemis to make things right.
It could be that Artemis is influenced by a new power she has acquired. She might be exploring a new aspect of her divinity, like a darker side of the moon's power. This new power is too much for her to handle at first and when Apollo comes to check on her, she accidentally unleashes it on him. This would then lead to a story of Artemis trying to master this power so that she can never hurt Apollo again, and Apollo having to deal with the physical and emotional pain of being hurt by his sister whom he trusts so much.
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.