Let's consider the user stories.
User story 1. As a user, I would like to book an appointment.
It seems like in this use case there is no place for confirmation. If I booked an appointment via your webapp my use story is fulfilled. Why sending confirmation email then? Let's consider another user story.
User story 2 I want to make sure my appointment is booked properly:
- the resource/person is available
- I didn't make a mistake while typing my credentials
This user story can be fulfilled without an email. Display a confirmation stating the resource is available, ask people to check their credentials. Of course, this solution is more error prone than a confirmation email, but it is valid. Shall we continue? Let's consider couple more.
User story 3 I want to make sure I won't miss this appointment, so I would like to add it to my calendar.
User story 4 I want to review information about my appointment, e.g. name of the resource/person
Ok, email will work the best for that, but we can show a link to download .ics file to add event to the calendar. The event may contain information about appointment etc. It doesn't work well on all platforms, especially mobile, but it is a valid option.
Frankly, for stories 2-4 I had to challenge myself with a question "How can we avoid sending email?". Email looks like an easier solution for them, but you may try to go without it because there is yet another user story: "My mailbox is crammed with emails and I don't want to be spammed with another one".
Without knowing all details, I'm not able to prioritize these user stories, so I can't make final judgment, but I hope my thoughts help to evaluate your actual situation and find the right answer.