Can regular languages be Turing complete?
In short, the answer is yes.

But you're mixing two completely unrelated meanings of the term "language" (yes, this is confusing):


A set of strings. "Context-free language" means "a set of strings which can be recognized using a context-free grammar". 
A way of specifying a computation. "Turing-complete language" means "a way of specifying programs in which the Turing machine can be specified".


Note that you can talk about "the C++ language" from two completely unrelated viewpoints, using the two unrelated meanings of the word "language":


C++ as a set of strings which are legal according to the C++ grammar
C++ as a way of specifying programs.


The traits of "the C++ language" from these two viewpoints are unrelated.

More examples to help you separate these concepts:


The expression "[a-z]+@[a-z].[a-z]" describes a set of strings recognizable by finite automata, i.e. a regular language. However, it's just that - a set of strings: is not a way of specifying programs (unless you ascribe a way to interpret each such string as a program), so it does not make sense to talk about whether or not it is Turing-complete.
The language of flowcharts is a way of specifying programs; depending on the particular flavor of flowcharts, it may or may not be Turing-complete. However, flowcharts aren't strings, so it makes absolutely no sense to talk about flowcharts in the sense "language as a set of strings".