There is smtp code around for the ethernet shield. Once the connection is established, the protocol is the same despite the transport. Here is one of mine:
And once you get to the point where it will send, it is not guaranteed the email will be accepted by the receiving email server. Many use spam blockers that will refuse email from non-commercial ip sets. If you do not have a commercial account, and you get your ip by dhcp, chances are you will be rejected.