You can only delete a block from the list if it's not used in the current drawing, that means no copies or references to that block.
Then you can PURGE the block definition.
If the block is used (referenced) in the current drawing, you have to update the local definition of that block. The traditional way to do this is to have a dwg file (made with the command WBLOCK) and then use the command INSERT and Browse. You will get a notification, "Update? Yes/No".
I am not sure if designCenter can be used to update existing block definitions. Try to double click instead of drag and drop. I'm afraid you'll get a small note like "duplicate definition ignored".