Here is one way of doing it
# Hospitals = (US Population x % who get sick once per year x % who visit hospital) / (average hospital capacity per year x average hospital utilization rate %)
US population x % who get sick x % who visit hospital: You can segment this out by age bracket and assign higher sickness % and hospital visits % to the more elderly age brackets.
The key assumption here is patients only visit hospitals once. You can add some more assumptions for multiple visits if you like
Average hospital capacity x utilization rate: Assume each hospital has 150 beds, average stay time per patient is 2 weeks, average utilization is 80%. Hence, the hospital can service 3000 patients per year (150 x 25 x 80%)
Putting it all together: Just to illustrate the logic, if US population is 300m, 20% get sick once per year, 10% of that goes to hospital. Then there are 6m hospital visits per year (300m x 20% x 10%).Then each hospital can service 3000 patients.
That then gives you 2000 hospitals.
A quick sense check - 2000 hospitals implies 1 hospital every 150,000 people. I live in San Francisco - which has about 800,000 people and about 5-10 hospitals. So that is about in the ball bark. It might be a bit low because we did not count for people who visit hospitals more than once per year.
Hope that helps!