Signals Cross Correlation from Their Power Spectral Densities?
No,

For instance 2 instances of AWGN with the same STD have the same PSD while being totally uncorrelated (Moreover, Statistically Independent).

Also, pay attention that Correlation is not commutative:

$$ f \ast g \neq g \ast f \Leftrightarrow F \left( k \right) \overline{G \left( k \right)} \neq G \left( k \right) \overline{F \left( k \right)} $$