What supports the Rockies? What supports the Rocky Mountains in North America? Or put another way, Why are they there? and Why are they still there?

It might be tempting to think of compression, especially in an east-west direction. However at the south end, New Mexico is experiencing east-west extension (grabens including Rio Grande Rift Valley, plus associated volcanism).

Hot spot and/or mantle plume uplift doesn't appear to explain them either. Yes there's the Yellowstone Hotspot under NW Wyoming. Regardless of whether this is really caused by a mantle plume or not, the related volcanism does not correlate with the mountains.
Part of the current topography might be due to a hot upper mantle below the Rocky mountains. This can be determined by seismic tomography. The hot mantle creates isostatic uplift.

Just one of many pictures I found:

http://srl.geoscienceworld.org/content/83/1/23/F2.large.jpg