What chemical can be used for getting rid of hard water spots on cars?
Maybe the water spots are not limestone, chalk or gypsum, all of which are dissolvable in acid. Maybe they are mud spots, which could contain clays which adhere to the car paint. 

If a toilet bowl cleaner based on citric acid (stronger acid than acetic, also a chelator) or Coke (acidic, contains phosphate) do not work quickly (without damage), perhaps an alkaline dispersing agent (like a liquid laundry soap) plus some elbow grease could diminish the visibility of the spots. Dispersing agents for clays include phosphates and anionic polymers like polysilicates (but they are typically quite alkaline - need to check damage potential). 

Then, a good waterproofing (waxy) polish could make it easier to resist and remove the spots. Discovering where the spots come from (rain, splash from puddles, lawn sprayer, etc.) could help suggest what materials are present in the spots.