Ending the Diagnostic Odyssey Act of 2019 This bill allows state Medicaid programs to cover whole genome sequencing services for certain individuals. Specifically, states may cover such services for individuals under the age of 21 (or a lower age, if the state chooses) and for former foster youth under the age of 26 who (1) have been referred or admitted to an intensive care unit or seen by a medical specialist for a suspected genetic or undiagnosed disease, or (2) are suspected by a medical specialist to have a neonatal- or pediatric-onset genetic disease. The Centers for Medicare & Medicaid Services may award grants to assist states in developing plans to cover such services.