Phase information from product of complex conjugate transfer functions
The answer to your question is no.

The relationship you show is well known and used to relate the input to output of Gaussian stochastic processes through a linear filter. Any filter that has the same magnitude response $\mid H(\omega ) \mid$ will produce the same result. 

One can often infer a phase by assumptions on the filter such as linear phase or minimum phase but this isn’t  usually the case.  

If you want the actual phase, you need to compute the cross spectrum.