Restore Public Health Institution Trust Act of 2023 This bill requires the Government Accountability Office to report on public health messaging and other recommendations of the Centers for Disease Control and Prevention (CDC). The report must specifically address (1) the data used to make recommendations; (2) the effect of inconsistent messaging on the public's trust in the CDC, willingness to follow the CDC's COVID-19 guidance, and attitudes about immunizations both in general and for specific populations; and (3) the degree to which outside entities influenced CDC recommendations.