If the game has been patched and fully installed then the game should not be asking you to insert the disk. According to "patch.txt" from the Diablo 2 installed folder....
Q u o t e: If all required Diablo 2 '.MPQ' files are installed on the
hard drive, the game will no longer require the CD to play.
For users that originally performed a 'Full Installation'
and wish to run without the CD, all '.MPQ' files should
be copied from the Diablo 2 CDs to the Diablo 2 directory.
Most users will only need to copy D2Music.mpq from the
Diablo 2 Play CD and/or D2xMusic.mpq from the Lord of
Destruction CD. Mac users will need to copy these music
files and rename them to 'Diablo II Music' and
'Diablo II Expansion Music' respectively.
Anyone who did not perform a 'Full Installation' will need
to re-install from CD again to ultimately play without the CD.
In this case, a 'Full Installation' is required, followed by file
copy step noted above.
In a nutshell. Try copying all of the *.mpq files from your CDs directly into the D2 folder and then try running it. This should fix the "Insert CD" message. I hope that is gets the ball rolling for you.