Care Veterans Deserve Act of 2019 This bill expands and makes permanent the Veterans Choice Program. The program allows eligible veterans to receive health care in their communities rather than at a Department of Veterans Affairs (VA) facility when certain hardships exist (e.g., when a veteran resides more than 40 miles from a VA facility). The bill expands program eligibility to include veterans with a service-connected disability rated at 50% or more. Additionally, the bill requires the VA to contract with a national chain of walk-in clinics to provide hospital care and medical services in such clinics to veterans who are in the annual patient enrollment system. The clinics may not require a preauthorization or a co-payment for treatment. Under the bill, an authorized health care professional may provide treatment via telemedicine at any location in any state, regardless of where such health care professional or the patient is located. The bill requires the VA to extend the operating hours for VA pharmacies and medical facilities. Finally, the VA may provide for a nongovernmental hospital organization best-practices peer review of each VA medical center.