Convolution theorem for cross-correlation
To stick to proper notations, if $\mathcal F \left\{x[n]\right\}=X(\omega)$ and $\mathcal F \left\{h[n]\right\}=H(\omega)$, what you mean is the Fourier transform of a convolution gives the product of the transforms of each term:

$$
\mathcal F \left\{x[n] * h[n]\right\} = \mathcal F\left\{\sum_{k=0}^{\infty}h[k] x[n-k]\right\} =X(\omega)\cdot H(\omega)
$$

As already mentioned, there is a similar formulation for correlation, the correlation theorem. And again, the Fourier transform of the correlation:
$$
\mathcal F\left\{\mathrm{corr}\left(x[n],h[n]\right)\right\}=\mathcal F\left\{r_{xh}[n]\right\}= \mathcal F\left\{\sum_{k=0}^{\infty}h[k] x[n+k] \right\}= X(\omega)\cdot H^*(\omega)
$$

The product $X(\omega) H^*(\omega)$ is called cross-energy density  spectrum and noted $S_{xh}(\omega)$, and this is just $\mathcal F\left\{r_{xh}[n]\right\}$. The $^*$ sign is for conjugate. 

Note that your convolution/correlation are on discrete-time signals $x[n]$ and $h[n]$. So $X(\omega)$ and $H(\omega)$ are transforms of $x[n]$ and $h[n]$ respectively, and not of continuous-time signals $x(t)$ and $h(t)$.