Racehorse Doping Ban Act of 2019 This bill requires that there be an independent anti-doping organization with responsibility for ensuring the integrity and safety of horse races that are the subject of interstate off-track wagers. The bill prescribes conditions under which such organization may (1) suspend the period a person is excluded from participation, and (2) permit the use of furosemide in a horse participating in such a race during the two-year period following this bill's enactment. A host racing association may conduct a horse race that is the subject of an interstate off-track wager, and an interstate off-track wager may be accepted by an off-track betting system, only if consent is obtained from such organization. Such organization must ensure that all costs incurred in carrying out its duties are defrayed pursuant to agreements for such consent.