Yes, you can configure emails sent on the signing completed event by using merge fields designed to attach a document in the email body.
The merge field to use in an email body to accomplish this is:
[Attachment - Final Document]
for the signed document, and additional values that can be used are
[Attachment - Envelope Completion Report]
[Attachment - Original Document]
[Attachment - Document Completion Report]See documentation on merge fields here.