Argoon: GoG help support team advised me to try to play the game with administrator privileges and that solved the problem for me, hope it does for you.
Yes, they told me the same thing, and that works for me.
So, in order to fix this once and for all, I opened the game directory, located the "Run Call of Juarez" shortcut. Right clicked on it and opened the settings dialog. Switched to the Compatibility tab, and there switched on "Run as Administrator". Now I can also launch the game through GOG Galaxy again.
Strange, though, that this is needed for some people. Does this mean that everybody else runs their games as administrator all the time? (rethoric question, obvsly)