What do you mean "delete a conversation with Siri"? Unless you instructed Siri to dictate an email, iMessage, SMS message or a note of some sort, the conversation is NOT written down anywhere.
Siri works as doing or inquiring something by voice in what you ask it. As JustMe'D said, Unless you ask it to dictate something to be sent that is it. Ask it to make an appointment for the Doctor on a certain day and time on your native calendar and that is it. It will confirm and ask you if you want Siri to set it and that is the extent of it. Thank you for the inquiry though.