They do react but the problem is, the alcohol reforms.
The aqueous NaOH reacts with your alcohol to form C-O-Na+ and water. The ethoxide ion will react with... water reforming the alcohol. These are in equilibrium so effectively, the net reaction is no reaction because the ethanol reforms.
Also, the pKa of sodium hydroxide may not be high enough to deprotonate ethanol to form your ethoxide... However, I cannot find a decent pKa table to compare the pKa of ethanol with sodium hydroxide.