First, you conclusion doesn't follow. If the total resistance (internal plus external) is actually zero and you apply Ohm's law, the voltage across is zero for any finite current through.
But if the cell has zero internal resistance as you stipulate, the terminal voltage must be non-zero and so you arrive at a contradiction, e.g., 1.5V = 0V.
If instead one starts with the external resistance $R = 0$ and let the internal resistance go to zero $r \rightarrow 0$, one sees that the current goes to infinity $I \rightarrow \infty$.
But no physical cell can supply arbitrarily large current and so you conclude that no physical cell can have zero internal resistance, i.e., any physical cell has a maximum short-circuit current.
Further, any physical arrangement of a wire and cell will have inescapable inductance since a loop is formed through which magnetic flux can thread.
At the point that the total circuit resistance becomes insignificant compared to the circuit inductance, you more or less discard Ohm's law and look at the fact that the inductance and cell voltage limit the time rate of change of current.
Finally, there is also inescapable capacitance and radiation resistance that might need to be considered.