We can use the narrowband spectrogram for finding the pitch of an utterance. But is it possible to find the pitch using the wideband spectrogram? If so, how would that be done?
If your window length is shorter than the pitch period of a voiced utterance, the spectrogram will not be able to capture the fundamental frequency. This is the problem, yes?
But evidence of the fundamental frequency will still be available in the spectrogram because spoken voice has plenty of harmonic content. If you can identify the upper harmonics, showing up as horizontal stripes in the spectrogram, and find the size of the gap between them, this will be the fundamental pitch. Say your FFT resolution is very coarse under 200 hz, but you find a peak at 240 hz, 360 hz, and 480 hz. Well, this is a dead giveaway that the fundamental was 120 hz.
Moreover: The glottal impulse (vocal folds slamming together) is like a broadband impulse. You will get periodic vertical stripes along the spectrogram if the pitch is low enough, and your FFT window short enough. The time difference between these stripes is the pitch period of the voice, so this is another clue you can use to determine the pitch even when the fundamental is ambiguous.
Frequency is the derivative of phase. Most spectrograms throw away phase, and just show spectral magnitudes. Wide-band spectrograms throw away phase more often, and thus are more informationally lossy regarding frequency information, compared with narrowband (longer time window) spectrograms.
One might be able to use non-local information (the overtone series or adjacents time points in the spectrogram) to infer more information about a particular frequency band, but this assumes the overtones are harmonically related and not inharmonic or accidental, or that the spectrum is stationary or predictable across time. If not, these estimation methods can fail.