It doesn't seem like that to me. Cellular automata are discrete systems with a quantized length scale (the cell size). By nature they can't exhibit the scale invariance that is the hallmark of a fractal.
At a sub-nuclear level? Perhaps I'm confusing a concept of summed structure.
As with all discrete systems, celluar automata can be made to approximate continuous structures.
Celluar automata is a kind of discrete dynamical system which is governed by a set of logical rules rather than continous functions, as in the case of a set of ordinary differential equations. Due to its discrete nature, it was a popular topic in Computer Science. Also in some physics circle.
I have to say that making things discrete frequently robs a system of its essential nature, while simplifying the system. Sometimes wrong simplication can lead to the wrong track. The essence can be left out during the simplification process, the process of discretization.