Ice Diamond Riddle SOLUTION ft. Vsauce's Michael Stevens

In this case (being contrary to Michael), I think you don't have to exaggerate things to a major scale.
It turns out so easy when you think this way:
Imagine the Ice cube is small, and the diamond inside is big, such that is almost diamond (diamond cover by an small amount of ice). This would have almost the density of the diamond, then it will sink instead of floating. Now it's all question of volumes: diamond will remain the same whereas ice will melt.  Ice has a lower density than liquid water, meaning that when melted it will occupy less volume. So... the water level will drop
No matter how big is the ice and how small is the diamond, the result is the same, doesn't matter if floats or not.