There can be 2 reasons for your native calendar on iDevices not updating automatically.
1. iCal feeds are not updated in real-time but there is a "Auto-Refresh" time that needs to be setup. By default, the time on mobile iOS devices are 5 minutes while in MAC you can control the "Auto Refresh" time. Here is an article that explains setting "Auto Refresh" time on MAC http://support.apple.com/kb/HT5029.
2. You might be using iCloud to synchronize Appointy iCal calendar on all your iOS devices. Make sure to subscribe to iCal on your MAC first and then subscribe Appointy calendar on MAC to iCloud. Subscribing first on mobile iDevices does not work.
Should you need any further help, please feel free to schedule a time with our technical team from https://support.appointy.com