The voltage is not simply something in a long list of 'must be present' in order for current to flow. Voltage _must_ be part of the injury process.
As I have said before, focusing on the current alone is a useful approximation simply because it appears that the level of injury is proportional to the current level, at least for many electric shock scenarios. Because the level of injury is correlated with the level of current flow, IMHO it is a reasonable shorthand to say that the current is causing the injury.
However a little thought will show that the above must be an approximate model. Most likely, different shocks of the same current but different voltage are in fact different, but that the effect of voltage is smaller than the effect of current...and with our current level of knowledge we can't really discern the magnitude of the voltage factor. Add to this the fact of physiological variation, and it might simply be that the voltage factor is smaller than the noise of different people responding differently to the same level of shock.
Just because the research has not teased out the relative importance of voltage level in determining shock injury does not mean that voltage is not a factor; just not a factor that we can usefully evaluate.
I can come up with two plausible examples where voltage level would make a difference given the same current level for a shock. These are just guesses, not known injury mechanisms:
For example, one mechanism of injury is disruption of electrochemical signaling in the heart. If you place electrodes right on the heart then a very low voltage will permit lots of current to flow and cause injury. But this voltage must be high enough to overcome the various electrochemical potentials (cell wall voltages and the like) involved. Perhaps with really good electrodes and a really low voltage, you could get 100mA to flow through the heart without actually flowing on nerves. (Likely, no, but plausible!)
On the other end of the spectrum, if you reach the point where thermal injury is a concern, then very clearly a higher voltage shock of the same current will deliver more power. In other words, if the injury is by cooking rather than confusing nerves, it seems to me that watts will be a more important factor than amps.
-Jon