When there is no resistance, as is the case with an ideal wire, any value of current satisfies Ohm's Law:
$V = I R$
since both $V=0$ and $R=0$.
But isn't V is like what causes the current?
Perhaps a mechanical analogy of the resistor will help. Consider the dashpot where the velocity of the arm is analogous to current while the force acting on the arm is analogous to voltage.
The relationship between the force and velocity for a dashpot with impedance $\mu$ is:
$F = \mu v $
This has the form of Ohm's law and is in fact its mechanical analog.
If the dashpot impedance is zero, the arm can have any velocity even though the force is zero. Physically, this seems reasonable since, when there is no external or damping force acting on the arm, we expect that the motion will be unchanging.
Similarly, if there is a steady current through a zero resistance (an ideal wire), we shouldn't expect that a voltage is required to maintain that current, we should expect that the current will be unchanging.