According to http://www.caverbob.com/wdeep.htm and http://www.caverbob.com/usadeep.htm, the only cave over 1000m deep in the US is a lava cave in Hawaii. In the continental US the deepest is 505m.
On the other hand in Europe there are two caves over 2000m deep.
The largest cave systems are in the US (Mammoth Cave in Kentucky. Why aren't they deeper?
It seems that the limestone layers are thinner and have been tilted less than in Europe. If this is the case, why?
It is related with the large-scale geology of the USA and Europe. The USA is rather normal in this regard, but the particular geological history of Europe allowed the formation of exceptionally deep caves there.
During Mesozoic, Europe was a closed ocean: the Tethys Ocean, with shallow carbonated platforms. The Alpine Orogeny formed the continent and the carbonaceous materials emerged creating mountains with large carbonaceous units. The meteorization and erosion of these carbonates created big karstic systems with deep caves.
Conversely, the USA has a lot of Paleozoic units where the carbonaceous materials are too compact to form deep caves. You may have shallow Mesozoic caves near the surface but Mesozoic units are thinner than some folded zones of Europe.
Deep caves primarily need two things: 1) Very thick or extensive tilted limestone beds so that there is enough bedrock to form a deep cave, and 2) A deep water table so that caves can form by dissolution of the limestone.
Although phreatic caves form beneath the water table, they seldom form to great depth because the water becomes saturated with calcium carbonate early in the groundwater flow system.
A deep water table is promoted by uplift or by incising of rivers into plateaus. There are few areas in the US with these conditions as noted in the other answer.
