That's an interesting argument, but the Civil Rights Act of 1964 and the Americans with Disabilities Act both define "public accommodations" as including certain private establishments, including the RV park and swimming pool in question.
The crux of my arguement was that it is not unreasonable to ask the guardians of a sick child for a doctor’s note to ensure the safety of the other patrons. Doing so is not a violation of the American’s with Disabilities Act. The guardians of this child have a very weak case for a lawsuit based on the information provided in the article.