A CNN can approximate a function on a fixed number of input variables, say $n$ of them. The set of functions on $n$ input variables isn't "Turing-complete". For instance, a boolean function $f:\{0,1\}^n \to \{0,1\}$ is always computable, as it can be computed by a program that just hardcodes the truth-table of $f$; and the set of such functions is not "Turing-complete".
Complication: CNN's actually approximate continuous functions $f:\mathbb{R}^n \to \mathbb{R}$... but a similar point remains (at least if the input is bounded).
Turing-completeness isn't really connected to universal approximation. For one thing: Turing-completeness talks about languages, which are subsets of $\{0,1\}^*$ and thus refers to discrete entities. Universal approximations talks about functions $f:\mathbb{R}^n \to \mathbb{R}$, and thus refers to continuous entities.
To qualify for "universal approximation", it's enough to be able to approximate all functions of $n$ variables (for each function, there exists a neural network that approximates it), so it talks about functions on inputs of bounded length. Turing-completeness requires the ability to compute all computable functions, which is a set of functions that has no fixed upper limit on the number of variables, i.e., it is a set of functions on inputs of unbounded length. Universal approximation could thus, in some sense, be considered "weaker" than Turing-completeness (though strictly speaking they are incomparable; neither implies the other).
