I think it's by default enabled in Nagios but.... myabe it's not the same situation with your config. Check if you have $HOSTOUTPUT$ and $SERVICEOUTPUT$ macros in host-notify-by-email and notify-by-email in commands.cfg?
Anyway if you don't, yes, it's possible. In commands.cfg file you can format the email message. Figure out where you want the required information and then add:
Status information: $SERVICEOUTPUT$
- this is how you will get the latest check output of a service
Status information: $HOSTOUTPUT$
- this is how you will get the latest check output of a host
The complete macro definition list is here: