There are only 900,000 some odd hospital beds in the whole USA, 330,000,000 people. At any given moment, about 2/3 are occupied. This could rapidly become a problem.
330,000,000 people aren't going to get COVID-19 all at once and of those who do get it, only an estimated 20% will have complications severe enough to potentially require hospitalization.
Also, you need to look at why all those beds are currently occupied. When I worked in a hospital, the unit that had the largest number of patients consistently was orthopedic. In fact, it was the only unit that constantly had to go into overflow. While some of those patients in that unit were there because they ended up with emergency surgery, typically patients were there due to elective surgery (join replacements mostly). Assuming a large number of people need hospitalization over COVID-19, I assume electric surgeries will stop.
A triage process will also happen that discharges patients who are able to take care of themselves at home sooner.
Finally, there are many hospitals that have mobile units that can expand their capacity. At the health system I work at, we have negative air pressure tents that can be set up in a day. We last used them during the H1N1 outbreak and they can house several patients at once. I know other hospitals in the area have similar things too. On top of that, there are many disused rooms in a hospital because they can't legally be "bedded". However, in an emergency situation, this does change. Plus hospitals can convert pretty much any single occupancy room into a double occupancy by simply rolling a bed in.
Due to this, the number of 924,000 represents licensed hospital beds in the US. The actual number of hospitals can hold is significantly larger, probably close to double that.
The biggest issue though isn't where to put patients, it's having enough staff to take care of them. We have a nurse shortage in the US and smaller health systems will run into problems having adequate staffing. We do have CNAs, MA, LPN, etc. so that does help, but still, we could see nurses working some really long hours if the virus ramps up hospitalizations. There's also funding to worry about, with all those nurses working in presumably overtime conditions, the amount for payroll will go way up too. I suspect the state or federal government will step in here though.
I'm not talking about the sky falling. I'm talking about hospital beds and math, I hope realistically. At the current understood doubling rate of infection, and currently understood rate of hospitalizations, the US may be running short of beds sometime in about two or three months, I figure.
Two or three months? I highly doubt that. A year of steady growth of the virus? Sure, that's a possibility.