I have calibrated several different monitors. Will Photoshop be able to pull up the correct color profile depending on which monitor I am showing Photoshop on?
I don't really want to provide 'half an answer' but I don't want to leave it as a comment either - so this is ½ an answer...
First of all, I agree with Rafael - the OS handles the profiling. So long as your display's profile is in the list of available profiles in Photoshop and is not selected as part of your workflow then the OS will handle it.
afaik it depends on whether you're on Mac or Windows & how many monitors are connected simultaneously as to how well it handles it.
To my understanding, Mac can automatically detect & follow all screens & compensate for each as you drag an app or image to it.
Windows cannot apply correct profiling for more than the primary screen.
You can test this empirically on Mac by dragging an image across between screens in Photoshop & watching it change as it passes half its area to the other screen. As it passes the line the colour looks "wrong" until you release the mouse, then it flips to the new profile. You can see this effect in Photoshop if you float a window free of the interface & drag it between screens.
The Finder [using QuickLook] does not display the same 'double compensation' it does it smoothly & you cannot see the change, the image continues to match perfectly at all stages.
 A common error is to set your display as the "working profile".
Never do this. Use sRGB, Adobe RGB 98 etc... as appropriate.
In my modest understanding of the color profiles, the monitor profile is saved in the operating system and used to control the values of the graphics card.
So there is no need for Photoshop to pull any profile for the monitor because it is assuming the output is defined by the OS.