It may surprise you, but the population density of Tennessee is 67/km^2, which sure, is a lot lower than Denmark, but also a lot higher than 36. Most of the east coast has a higher population density than Denmark. Florida has a higher population density than Denmark. When people talk about the low population density of the US, they're averaging in a lot of sparsely populated land west of the Mississippi, which has fuck-all to do with whether rail lines would be useful in the parts of the US where people actually live.
Florida has a high speed rail that runs between Miami and Orlando and has plans to expand. It's called Brightline, and it's one of the few successful passenger rail projects in recent years in the U.S.
Also population per km^2 can be incredibly misleading, because it's an average. One example could have the population far more evenly spread across the land area, and another could have a dense urban core surrounded by farmland or national parks.
Florida has a slightly higher population than Denmark, but as an example, one of it's largest cities, Orlando, has a population density of ~283 people / km2, while Copenhagen has a population density of 6800 people / km2.
I have lived in Florida all my life. It is sadly the definition of sprawl. There is no way to build enough rail that would serve the average Floridian for day to day transport.
