Why does magnesium oxide not react with water?
Like a lot of things in chemistry, "it depends".

Magnesium oxide is commonly produced by calcination of magnesium hydroxide or carbonate. The magnesium hydroxide or carbonate decomposes at a fairly low temperature whereas magnesium oxide is itself stable to both decomposition and melting  beyond 2000°C. So, we can choose from a wide range of calcining temperature, and as explained here we can use this to dial in the reactivity we want. 

At a low calcining temperature the magnesia is quite reactive, even hygroscopic (the "magnesium oxide" film formed on magnesium in ambient air is thus usually magnesium hydroxide); this "light-burned magnesia" is used where easy reactivity is required as in water or soil treatment.  And, yes, the reaction with water gives magnesium hydroxide.

Calcination at higher temperature gives less reactive "hard-burned" magnesia, or in the extreme case the totally unreactive "dead-burned magnesia" used in furnace refractories.