I think it can... sort of. I can feed test signals of some sort into an amp when driving a test load, and I can record what happens to the signal. But here are the caveats:
1. The amp's 'signature' that the fans love so much is inextricably linked with its dynamic interaction with the speaker. This includes what the amp does when the volume is low, when high, what it does with transients. The voltage and/or current we record is only 'valid' with that speaker after its voice coil has warmed to the temperature it is during "the sweep", the ambient temperature affecting the speaker's mechanical parameters and so on. Feed the same voltages into a different speaker, or even the same one at a different temperature, and the result won't replicate what the amp would do with the new speaker. We could test at all temperatures with all models of speaker and then feed in the ambient temperature, voice coil temp etc. into the model during playback, but is this not getting out of hand somewhat? Also, simple tolerances and variations between the same model of speaker would invalidate our model.
2. No "sweep" can record everything about the amplifier. It will need to be more than just steady tones, or white noise etc. The result cannot just be a simple 2D lookup table, but must incorporate a window of previous samples in order to approximate the dynamic behaviour of the amplifier interpolated from, effectively, a hypersurface.
3. I think that with valve amps there is another factor: microphony, that is room dependent and so cannot practically be duplicated in a software simulator.