How old is your version of Diablo? I'm not sure how backwards compatible it might be, but it could be a reason.
Are there any available updates for Diablo that you can download? I know my Diablo II wasn't working properly when I installed it first time (I know you're on about Diablo I though) and when I downloaded and installed an update it worked. Latest one can be found here.
Have you read the Diablo FAQ/Technical Support? Blizzard have a Technical Support area for problems. I wouldn't think yours is unique, so it's worth a look. I did just take a small glimpse over at it and found a problem sounding like yours here. Might wanna take a look.
I hope that helps and your game gets working soon.
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are you using dual monitors? I couldent get D1 to work for the life of me. installed and ran fine on my laptop, but then wouldent work on my new comp. then I disabled dualview and it worked just fine. also had to make an exception for Data Execution Prevention for it to run as well. I should probably go back and enable dualview with the DEP exception to see if it will still work that way.