why was lily so insistent that rosaleen leave the hospital?

Lily doesn't want anything to happen to Rosaleen as she's all she has. The odds of the men who beat her up (racists) getting in trouble were nil and none; she knew that the one who'd get in trouble or be hurt further would be Rosaleen.

"The only memory that Lily has of her mother is "blurry" and takes place on the day her mother died, back when Lily was only four years old. One day, while in town with Lily, Rosaleen insults three of the Town's racists. She gets arrested and ends up in the hospital with injuries. Lily breaks Rosaleen out and they run away to Tiburon, South Carolina where they meet and move in with the Boatwright sisters. The story is about how Lily finds her true home and experiences her first love."