I have a backsplash that is made of a sheet of formica. I want to tile my backsplash, but can't remove the formica and don't want to add a 1/4 inch thick layer of hardibacker.
Can I rough up the formica with 60 grit sandpaper (or rougher) and then apply mastic and tile? Will it hold?

Something is missing in your explanation. The formica is about 1/16 " thick and is usually mounted on MDF of other hard surface material to give it thickness of heft. You can split the formica from the MDF with a knife or chisel. Tile can then be attached with modified thinset and grouted.

The short answer is no. not with ceramic tile. However you can make, or have made, tiles out of a solid surface material and adhere them with 100% silicone. You would tightly but them and cut off the bead that oozes out. An advantage to this is no grout line to clean.

I like ballengerb1's answer better.

To respond to ballengerb1: true... formica is usually attached to some type of backer to give it some heft... however in this case the extremely frugal person who built the house I now own put the 1/16" sheet of formica directly on the wall attached to the drywall with some type of adhesive (actually seems to be contact cement). I tried to take it off, but that will entail cutting around it and tearing out the drywall and all..not something I really want to do. It looks like my only recourse is to put up 1/4" hardibacker and deal with making the raw edges of tile and backer look finished... probably a trim of some sort.

OK here are two thoughts. The formica can probably be split from the wall with a stout putty knife, don't woory about the drywall paper since it will be covered. There is absolutely no way that the formica won't come off with this method, the drywall paper has only a small amount of strength and it will detach. Your edge can be accomplished with a tile bullnose or a trim piece manufactured to match your tile.

Just put the stone over the formica I did it looks great.