Are all your Shortcuts missing the Compatibility Tab?
Is the tab available on the actual .exe that the Shortcut points to?
Is this a 64-bit machine? Is Diablo a 16-bit program?
List of limitations in 64-Bit Windows
http://support.microsoft.com/default.aspx/kb/282423#appliesto
If you type "compatibility" in the Start Menu SEARCH box are you able to launch the Compatibility Wizard and use that as a workaround?
Have you tried creating a new shortcut to the program?
Try running ChkDsk to check your drive for errors. Right-click your Drive icon/ Properties/ Tools/ Error Checking. Try it first by not checking either box (this will run it in a Read-only mode) to see if it flags any hard drive or file problems. If it does, restart it by ticking both boxes, and rerun it to allow it to attempt to fix any found problems.
If you decide to reinstall Diablo look for the Compatibility Tab on the setup.exe and also try that and try this.
Try right-clicking on the actual Setup.exe and selecting the "Run As Administrator" option? Do so even if your user is an Admin.
Have you tried right-clicking on the shortcut to the program, or the .exe, and selecting the "Run As Administrator" option? Do so even if your user is an Admin.
Windows Virtual PC
Download Windows XP Mode
http://www.microsoft.com/windows/virtual-pc/download.aspx
Windows XP Mode (Windows 7 only, but not any of the Home versions) "You are not eligible to download Windows XP Mode. You must have Windows 7 Professional, Enterprise, or Ultimate to run Windows XP Mode". If you have an old XP CD (or other older Windows CD) available you can just download the Virtual Machine and skip the XP Mode download.
This covers the whole gambit of virtual machines in "Home" versions if you want to have a look at it. Of course you will need a licensed copy of XP to install and run in any of them (except XP Mode).
VirtualBox is an impressive VM contender