FFT of audio, packet sampling and others
As some comments have pointed out, there are a couple of practical ways to find out the sampling rate:


Read the sampling rate information from the WAV file header
Look for mains hum in the recorded audio, which you'd see as a peak in the FFT response in the frequency bin corresponding to 60 Hz (or maybe 50 Hz).


But it sounds like you're looking for a mathematical property of the FFT that will allow you to back out the sampling frequency. Such a thing doesn't exist. 

The only inputs to the FFT are the original sampled data. And this sample data itself has no information about the real sample rate contained in it. So there's no possibility that the spectral result of the FFT can tell you anything about the sample rate.

For example,  there is no way to tell the difference between a 10 Hz signal sampled at 100 Sa/s from a 320 MHz signal sampled at 3.2 GSa/s.

In order to know the sample rate you must record it as meta-data, for example in the header of a WAV file.


  If I have a .wav file with N datapoints then split into ten N/10 segments and then fft on each segment. But this then means that my sampling frequency is essentially reduced to N/10 Fs?


No, the sample rate is still Fs. But the sample duration has been reduced to N/(10 * Fs). 

This means that the spectrum will still cover the same frequency band (0 to Fs*(N-1)/N), but it will cover it in coarser steps. The bins of the DFT will be spaced by 10*Fs/N instead of Fs/N.