I am not a CS expert, but this does not look like a full implementation of rule 110, nor is it even pure CSS (there is HTML involved).
What I see in the SO answer is an interface for Rule 110 with an additional set of instruction (written in a natural language) for the user to execute manually. So you can use CSS + HTML to create an interface for a Rule 110, which is then written in a natural language around that interface. The answer even states that (very relevant) caveat.
> [...] so long as you consider an appropriate accompanying HTML file and user interactions to be part of the “execution” of CSS.
> The formal definition (simplest) of Turing Machine is simply a tuple of states set, symbol set, initial state, accepting states set and a transition function. There is no crank in it. By computation we mean somebody needs to apply the transition function faithfully on the tape which is exactly like the clicking in this case. More formally, a model of computation can be viewed as a set of rules somebody needs to follow to do the computation. In that sense, I think CSS is Turing-Complete.
What I see in the SO answer is an interface for Rule 110 with an additional set of instruction (written in a natural language) for the user to execute manually. So you can use CSS + HTML to create an interface for a Rule 110, which is then written in a natural language around that interface. The answer even states that (very relevant) caveat.
> [...] so long as you consider an appropriate accompanying HTML file and user interactions to be part of the “execution” of CSS.