State Accountability, Flexibility, and Equity for Hospitals Act of 2023 or the SAFE Hospitals Act of 2023 This bill alters Medicaid requirements relating to payment for inpatient hospital services that are provided by disproportionate share hospitals (DSHs). (DSHs are hospitals that receive additional payment under Medicaid for treating a large share of low-income patients.) Among other changes, the bill requires state Medicaid programs to adopt a payment methodology that meets certain criteria, including by prioritizing payments based on the DSH tier for which the hospital qualifies; tiers are determined based on factors such as the hospital's Medicaid inpatient utilization rate. The bill also incorporates state poverty ratios (i.e., the number of qualifying low-income individuals in a state compared to all states) into the formula for determining state DSH allotments under Medicaid. The bill phases in application of the revised formula over the course of 10 to 15 years.