I think Mark Hovey has pointed out the remark necessary to finish the proof. If we work in the **injective** model structure, then being fibrant is equivalent to being injective. If S are the stable equivalences then the S-local objects are necessarily injective spectra. Now use lemma 3.1.5 and a generalization of example 3.1.10 to proof that injective Omega-spectra are all S-local objects.

I therefore think we can conclude: the stable model structure on symmetric spectra is the left Bousfield localization of the injective model structure on symmetric spectra with respect to the stable equivalences.