How to estimate radial distortion from lens characteristics? Is there a way to estimate the radial distortion coefficients without calibration, given the lens characteristics, such as its diameter and its focal length?

Typically for camera calibration a pinhole camera model is used, and then radial distortion is added on top of that. Radial distortion is a 4th or 6th degree polynomial with 2 or 3 coefficients. During calibration, these coefficients are not derived from anything. They are initially assumed to be 0, and then estimated from data using numeric optimization.

What I am asking is whether or not there is an equation relating lens characteristics that you can read off the manufacturer's spec (focal length, aperture, field of view) to the radial distortion coefficients from the typical polynomial distortion model? In other words, is it possible to get some approximation of the distortion coefficients from the camera spec, before doing the calibration.
There is a lot of variability because distortion can be compensated for optically and lens designs differ. Some lenses are marketed as rectilinear, most not. You would be making the distortion worse for some lenses by doing any correction advised by focal length alone. For focal lengths shorter than about 30 mm, the distortion seems dominantly barrel, according to the data of:

Neale, W., Hessel, D., and Terpstra, T., "Photogrammetric Measurement Error Associated with Lens Distortion," SAE Technical Paper 2011-01-0286, 2011, doi:10.4271/2011-01-0286.

They only tested zoom lengths though.