@BitFlipper:

When I drag a URL to the desktop, the shortcut always brings up the 32-bit version. I checked this on both a Windows 7 Pro machine and a Windows 7 Ultimate machine.

I checked the Internet Options ' Programs dialog, and it says "Internet Explorer (64-bit) cannot be the default browser". 

The 64-bit version shouldn't even be allowed to be the default.

What I would do is set a different browser (e.g. Chrome or FireFox) as default, and then set IE back as the default to see if that fixes the problem.