A regular expression is by definition a regular grammar, the least mighty language in the chomsky hierarchy and realizable by finite state automaton.

You can get something similar to a virtual equivalent of a turing complete language, due to the fact that the tape of the turing machine is, contrary to the hypothetical theory, practically limit by recourses. CPUs are finite state automatons, yet we use them to solve problems of sufficiently low complexity in NP space.

I am not an expert, please correct me if I am wrong.