Is double-click working normally everywhere else? i.e. it's definitely not just the mouse button ?
Also try highlighting the icon with a single click, then pressing enter. Does that still work or not? Enter is supposed to equal double-click and failure here might help diagnose the problem.
Sometimes the desktop icon cache becomes corrupted. To rebuild it,
Right-click dektop, select properties.
Go to settings tab->Adjust font size dpi.
Change your font size to 120 dpi and reboot the computer.
Again change the font size to 96 dpi and reboot your computer again.
There are several other ways to rebuild the cache - we'll try those if this one does not work.