How can I update Apple Account email on my Apple Watch?

I received a new Apple Watch recently. When I sign into it, I receive the message "Enter the password for your Apple Account to continue using Face-Time on this Apple Watch: ". The email it listed for my Apple Account sign in here is outdated however, since I had the email address deleted in the past and switched to a Gmail account. My current Apple account has the new email instead of the old one, so I am confused why it is showing the old one on this Apple Watch message, and I am unsure how to change it so I can sign in to the new email instead. Does anyone have any advice?

[Re-Titled by Moderator]