And no romex. There are a lot of warning signs in your post. You will need a permit and inspection for this and it should be done by a licensed electrician.

See table 300.5 of the 2016 California electrical code for your answer. Ron is correct, there is no depth requirement under a building for your scenario. Your electrician is not competent.

Also in California, you are generally not required to use a licensed electrician if you are pulling an owner-builder permit. For future electrical questions refer to the version of the CEC your city has adopted. That will be your bible when it comes to electrical. Good luck!

18" is for burial OUTSIDE underground. Inside the structure, you can place it anywhere it will be relatively safe (and Sched80 PVC will do nicely in a poured floor).
20A requires 3 12g copper wires. Technically, you can use 1/2" conduit here, though it's usually easier to work with 3/4".

The protection against physical damage is in Article 304 of the NEC (specifically paragraph (F)); He seems to want to apply Article 305 which is Underground which doesn't really apply here.

For interior applications there is no minimum depth requirement, the conduit provides the protection. If you are at all concerned then use metal conduit versus the PVC version. Your electrician is likely referencing the depth that would be required when running conduit underground in an exterior application.

Why no Romex? If your wiring inside a wall is going through conduit to a cabinet, then to avoid Romex in the conduit, you would have to have two junction boxes to connect the wall Romex and the cabinet Romex to the wires in the conduit. Of course, you could use flexible conduit ("Greenfield") in the wall and cabinet and pull individual conductors, without having extra junctions, but that may be extra work. Thoughts?

That's not an electrical requirement. You shouldn't be willy nilly "trenching" through such a slab. Going down 18" isn't going to solve that.

Romex here iffy. Generally, you do not put type NM in conduit, but it is permitted when done to "prevent physical damage." No big thing putting a junction box on the end (it likely ends up inside cabinets in either case). If you're going to run NM, you better go even larger with the trade size of the conduit. It's a bear pulling it more than a foot or two in a minimally sized conduit.

Any elecrician worth his salt can wire up the boxes in the island without having to go back to cable or transitioning to flexible conduit.

Just to clarify the reference to Romex. Romex is a brand/trade name for NM-B, which often refers to 12/2 or 14/2 cable that is jacketed with two hot wires and a ground inside the jacket. Its also available with a third conductor as 12/3 or 14/3.

Although NM-B could be run in conduit under the slab, the outer jacket makes it larger and harder to pull. The more common solution is to pull three separate THHN or THWN conductors inside the conduit, since it is much easier to pull through the conduit. Pulling NM-B through conduit corners and angles is almost impossible unless the conduit is much larger than the cable, since the jacket binds as it bends around the corner. Your electrician should not have any code issues with either solution.

Bruce