Why is my ceramic tile popping out of the floor?
This is a really really poor installation.  Everything will have to be pulled up.  Tiles will need to be cleaned or thrown away and reinstall.

Issues:


The mortar/cement isn't even covering under the mesh in some pictures.  To do this type of install you need to cement/mortar the mesh to the floor and a skim coat on top.  Once this dries you can continue the tile installation with thinset.
Either the mix wasn't right or the contractor used the mortar after it started drying.  I can tell by the pictures that this wasn't a good wet install.
The tiles were just thrown on top of the mortar after he added it to the mesh.  In some areas there was not enough mortar so the mortar seeped under the mesh and the tile had nothing left.  (and I am 80% positive that the mortar used for the mesh is probably not good for tile install but let's just assume it is)
There is no sign of a trowel being used (properly).
And then the biggest reason... It is completely obvious that the tiles were added way too late.  The tiles look like they were laid on mortar that was 70% dried out.  You are lucky about this because this floor was going to fail miserably eventually and you would be out more time and tiles.  But this screw up will actually make demoing and reusing the tiles easier.  


This is just a guess.  But it looks like this guy laid out the mesh, nailed it down, mixed up a couple of big buckets of mortar and just threw it on the floor.  Once he "smoothed" out the floor he started adding the tiles.  That is why you are seeing so much inconsistency - some mortar sticking to tiles, some to floor and some in between.  

My advice is to demo everything that is left and hire someone that knows what they are doing - I am happy to give you step by step advice as this is much simpler than this guy made it.    You cannot let this guy touch your house again though.