Cross Power Spectrum
Whenever talking about Cross Correlation of signals it is important to well define it.  

If you mean Cross Correlation in most common sense (Signal Processing):

$$ \left( x \star y \right) \left[ n \right] = \sum_{i = 0}^{N - m -1 } {x} \left[ m \right] y \left[ n + m \right] $$

Then of course not.
You can have cases where the DFT of the Cross Correlation has higher absolute values then the DFT of the Auto Correlation.


  The Schwarz inequality answers your question, Yes.
  But don't use pwelch for the cross spectrum. The cross spectrum is complex valued.


The above by @Stanley Pawlukiewicz is wrong.
Simple way to see so is by defining $ y \left[ n \right] = 10 x \left[ n \right] $ which of course mean the cross correlation between those is higher than the cross correlation between $ x $ and itself.  

The reason is that Cauchy Schwarz Inequality is about Inner Product Space.
So if we talking about signals as vectors then we need to use the Inner Product which is the cross correlation at $ 0 $.
If we talk about Random Process than we need to use their definition (Which normalize the energy and then everything works).

If you use Cross Spectrum while sticking to the Random Process definition and normalize the Cross Correlation by the variance of the processes than indeed by Cauchy Schwarz Inequality you're guaranteed to see what you got.