Fairness to Pet Owners Act of 2019 This bill requires veterinarians to provide pet owners with a copy of animal drug prescriptions prior to filling or dispensing such prescription. Further, veterinarians prescribing animal drugs may not require a pet owner to (1) purchase the prescription as a condition of providing such a copy, (2) pay an additional fee to receive such a copy, or (3) waive the veterinarian's liability as a condition of providing such a copy. Veterinarians are not required to provide a copy of prescriptions administered in the context of acute care. A veterinarian also may require payment of examination fees prior to prescribing an animal drug if such fees are required regardless of whether the diagnosis reveals an animal drug prescription is necessary.