This starts as a good bit of computer science nerdery, that kind of answers the question in the title:
Alone, CSS is not Turing complete. CSS plus HTML plus user input is Turing complete!
And so the takeaway here is bigger than just speculation about Turing completeness:
Given that CSS is a domain-specific language for styling user interface, this makes a lot of sense! CSS + HTML + Human = Turing complete.
At the end of that day, as CSS developers that is the language we really write. CSS is incomplete without HTML, and a styled interface is incomplete without a human to use it.