I have a partially sunken cultured marble bathtub with a six-inch high surrounding cultured marble step-over ledge. I want to place a porcelain tile mosiac on the sides of this ledge to match the bathroom floor.

I sanded the cultured marble ledge, applied TEC Double Heavy Duty Ceramic Tile Adhesive (sometimes referred to as "mastic"), and applied a few "test" tiles. The mastic around the edges of the test tiles hardened and cured fine. After 72 hours, however, I decided to remove the test tiles to see if the rest of the mastic had also cured. Unfortunately, the test tiles all removed with little resistance and I discovered that the mastic beyond the edges of the tile was in the same condition as when it was applied. I assume that the mastic at the edges cured because it was exposed to air, whereas the mastic between the nonpourus tile and the nonpourous cultured marble surface was, in effect, sealed from the air and neither curing nor adhering.

How do I solve this problem? ANY suggestions will be appreciated.

Given that you probably only have a small number of tiles to set I would use a silicone. Because the cultured marble and the tiles are both non porous both the mastic and the silicone will take some time to cure. I believe the silicone would cure faster and adhere better.
Suggest that you test the silicone but give it a couple days to dry.