Can I create a private calendar appointment with an iPhone?
Solution 1:
No you can not.
I assume you're using Exchange for you calendar, although you didn't specifically say so.
You can't necessarily create a private event, but you can create a subcalendar in Exchange that you can control the permissions of so that only you can see it. You can do this by:
Open Outlook
Go to Calendars
Right click on your current main calendar and click New Calendar...
Create the new calendar under your Exchange account's Calendar section
Right click on your new calendar and select Properties.
Go to the Permissions tab.
Make sure the "Default" user is set to None.
Set your own permissions to "Owner" and remove any other users from the list.
This calendar should show up on your iPhone fairly quickly. If it doesn't, try disabling Calendar syncing in Settings → Mail, Contacts, Calendars → Exchange.
Solution 2:
Apparently Apple quietly added the ability to mark calendar events as private in iOS 8. At the bottom of the event details there is the option to toggle an event as Private. Setting this toggle works with Exchange.
See screenshots below:
