Not according to the source I found on Nucleation. All it takes is one impurity. that crystal then "Build on each other". Notice also it says "Absence of impurities". Not absence of most impurities". Here is this on Supercooling with additional comments on the necessity of Pure water. And a bit on how that would be achieved.
"A liquid below its standard freezing point will crystallize in the presence of a seed crystal or nucleus around which a crystal structure can form creating a solid. However, lacking any such nuclei, the liquid phase can be maintained all the way down to the temperature at which crystal homogeneous nucleation occurs. Homogeneous nucleation can occur above the glass transition temperature, but if homogenous nucleation has not occurred above that temperature an amorphous (non-crystalline) solid will form.
Water normally freezes at 273.15 K (0 °C or 32 °F) however it can also be "supercooled" at standard pressure down to its crystal homogeneous nucleation at almost 224.8 K (−48.3 °C/−55 °F).  The process of supercooling requires that water be pure and free of nuclei such as having undergone reverse osmosis, however the cooling itself does not require any specialised technique. If water is cooled at a rate on the order of 106 K/s, the crystal nucleation can be avoided and water becomes a glass."
It is interesting how some individuals only want to see or hear that which agrees with them. Sort of like forming a theory and then gathering only the information that makes it look correct.