• It appears that there is no data or information that compares the number of hospitals built against that of prisons.

Number of Hospitals

• "The American Hospital Association (AHA) is the national organization that represents and serves all types of hospitals, health care networks, and their patients and communities." They conduct and publish an annual survey of hospitals in the US annually but these reports require premium access. However, they do provide a few historical data with the most recent being for 2018.
• In 2018, there were 6,146 hospitals in the US — of which 5,198 were community hospitals (non-federal), 209 were federal government hospitals and the rest being other non-federal types of hospitals.
• In 2017, there were 6,210 hospitals in the US — of which 5,262 were community hospitals (non-federal), 208 were federal government hospitals and the rest being other non-federal types of hospitals.
• Statista provides the total number of hospitals in the US from 1975 to 2017. Its 2017 data (6,210) corroborates that published by AHA. The CDC also provides similar historical data.

Number of Prisons

• The total number of prisons in the US aren't exact or precise, likely due to the various facility types (such as local jails, juvenile correctional facilities, immigration detention facilities, private detention centers, etc) that are available.
• According to the Federal Bureau of Prisons, they list that there are 110 federal prisons in the US.
• The Prison Pollicy Initiative is non-profit, non-partisan Prison Policy Initiative that produces cutting edge research around the US justice system.
• It reports that in 2020, "The American criminal justice system holds almost 2.3 million people in 1,833 state prisons, 110 federal prisons, 1,772 juvenile correctional facilities, 3,134 local jails, 218 immigration detention facilities, and 80 Indian Country jails as well as in military prisons, civil commitment centers, state psychiatric hospitals, and prisons in the U.S. territories."
• In 2019, it also reported that "The American criminal justice system held almost 2.3 million people in 1,719 state prisons, 109 federal prisons, 1,772 juvenile correctional facilities, 3,163 local jails, and 80 Indian Country jails as well as in military prisons, immigration detention facilities, civil commitment centers, state psychiatric hospitals, and prisons in the U.S. territories."