COVID-19 Safe Birthing Act This bill expands coverage of maternal health care under Medicare, Medicaid, the Children's Health Insurance Program (CHIP), and private health insurance. The bill specifically provides for Medicare and Medicaid coverage of maternal health services that are furnished via telehealth. It also requires Medicaid, CHIP, and private health insurance to cover prenatal services and one year of postpartum care without cost-sharing. The bill also requires short-term, limited-duration health insurance plans, association health plans, and health care sharing ministries to cover COVID-19 testing without cost-sharing. It also requires state Medicaid programs to provide coverage of COVID-19 testing for pregnant individuals who are uninsured. Additionally, as a condition of Medicare participation, hospitals must permit pregnant individuals to be accompanied by someone of their choosing (in addition to a doula or other health care worker) during labor, delivery, and recovery, and may not implement any discriminatory policies that restrict hospital access or birthing options for pregnant individuals.