Care Veterans Deserve Act of 2021 This bill expands the availability of care for veterans at Department of Veterans Affairs (VA) facilities and non-VA facilities. Specifically, the bill requires the VA to furnish hospital care, medical services, and extended care services via providers under the Veterans Community Care Program to veterans with a service-connected disability rated at 50% or more who elect to receive care or services through a non-VA provider. The bill requires the VA to extend the operating hours of its pharmacies. The VA must enter into contracts with physicians and nurses to work at VA medical facilities during nights and weekends. Finally, the bill authorizes the VA to provide for a best-practices peer review of each VA medical center by a nongovernmental hospital organization to evaluate the efficacy of the care provided at each center. VA medical centers with the longest wait times for appointments or the worst health outcomes shall be given priority for peer review.