I asked the question in the title in a paper that I deposited at the SSRN elibrary today.
The reverse question, i.e. whether the current stock returns are good predictors of future volatility, has been asked and studied many times in the literature.
However, the causality from current volatility to future stock returns did not attract the academic attention at the same rate (that is not to say this way of causation has never been studied. I am only saying this way of causality is much less questioned.)
This is interesting because, although the academic community does not pay too much attention to whether volatility predicts stock returns or not, investment community (aka market professionals) seem to believe that the VIX index (a volatility index) is a gauge for future movements of S&P500 (a stock exchange index).
Well, long story short what I did in this paper is I constructed a near-VAR system which allowed me to model the dynamic interdependencies between the variables (variables are the changes in the VIX implied volatility index - that is the volatility - and S&P500 returns - that is the stock returns).
Then I estimated the model and found that volatility (at least the implied volatility measure I used) could not predict the relevant stock exchange returns. Yikes!
See this paper on SSRN: