For the lightly damped system damp ratio is approx. 1/2Q, where Q = f(res) / (f(high) - f(low)).
f(high) and f(low) are the half power points. They are the points where the squared magnitude of the response (your FFT values) are one half of the peak value, i.e. the value at the resonance frequency f(res).
If the FFT array is called g, then you want amgSqrd_g = (abs(g))^2 and then find the max and the two points nearest max that are nearest to or below the half max value.
Try the attached vi. I haven't tested it, but it may provide a starting place for you.
jc
Mac 10.4
LV7.1
CLD