On my iPhone 4S with iOS 7, I have my default calendar set up as Gmail. This does sync my calendar just fine. I can create an event on the phone or the web and it syncs both ways.
The weird thing that I cannot figure out is: When someone adds me as an invite on their Google Calendar, I do get an email on the phone, but I DO NOT ever get a notification on the iPhone Calendar app. I've tried other Calendar apps, and it is the same behavior.
Any ideas on settings to check?