Can You Solve this Ice Diamond Riddle? ft. Vsauce’s Michael Stevens

The water level will go down.
Explanation: to simplify let's imagine that the diamond is suspended under the ice cube instead of being inside and the void inside is filled with water. To convince yourself that this is equivalent imagine a boat with a load inside. If you move the load under the boat and suspend it then the boat goes up because now there's some extra lift (equivalent to the weight of water of the volume of the load). So if you add the volume of water equal to the volume of the load inside then the boat goes back to its initial position.
Now back to the diamond. Now the diamond is not suspended anymore and sinks. The ice cube can go up a bit because it doesn't have to provide lift for the diamond.  The diamond that was already completely in water doesn't change anything. Conclusion the ice cube goes up and the water level goes down.
Finally the ice cube melts and that doesn't change anything.
So: level goes down.