By natural I mean that we could have simple laws of physics and initial conditions in which the creatures evolve over a long period of time, as we have seen in our universe.
It is very likely that even Conway's Life universe has this feature. Its rules are absurdly simple, and we know that it can contain self-replicating structures, which would be capable of mutation, and therefore evolution. We can specify very simple initial conditions from which self-replicating structures would be overwhelmingly likely to appear, as long as the lattice is big enough. (The binary digits of many easily-computable real numbers would work.) Moving from this 2D world, in which each cell can be pictured as a square with 4 orthogonal neighbors, we can consider 3D CA in which each cell is a cube with 6 orthogonal neighbors. There are rule sets and initial conditions for this lattice structure that are just as simple as Conway's life, which can similarly contain evolving self-replicating structures. We can go further and envision a 4D CA in which each cell is a hypercube with 8 orthogonal neighbors. Without a doubt, there are absurdly simple rulesets for this lattice structure which are computation universal, support stable structures like gliders, and support self-replicating structures which would grow and evolve.
Universes of the natural type would seem likely to have higher measure, because they are inherently simpler to specify.
If that's true, then the CA universes described above should have very high measure, because they are extremely simple to specify.
Tegmark goes into some detail on the problems with other than 3+1 dimensional space.
Once again, I don't see how these problems apply to 4D CA. His arguments are extremely physics-centric ones having to do with what happens when you tweak quantum-mechanical or string-theory models of our particular universe.