At room temperature water is more stable because ice will tend to melt at room temperature.
The change in entropy of the system during the phase change is, in this case, given by the heat absorbed by the ice divided by its temperature:
$\Delta S =\frac{ Q }{ T }=\frac{ mL }{ T }$ which is positive.
This shows that the entropy of an amount of ice at $0^{\circ} C$ is less than the entropy of the same amount (mass) of water at $0^{\circ} C$.