Is there any way to manually initiate a refresh in the Calendar app in iOS 7? (I'm using iCloud to sync my personal and shared calendars with two Macs and two other users.)
In the iOS Calendar app, at least between the introduction of iCloud and iOS 6, you could show the list of calendars and then tap the refresh icon (two arrows in a circle) at the bottom of the list. In many other apps you can do so by pulling the main list view down and releasing. But I have not found any way to force iOS 7 to do this. If there is no such mechanism, consider this a feature request.
"Wait until the app refreshes by itself" is not really a valid answer to this question. Sometimes it just doesn't, and a manual refresh might help break a log jam. And sometimes you're about to get on a plane and need to be sure your calendar is up to date before you go into Airplane Mode.
I think you can tap Calendars at the bottom, then pull down to refresh the Calendars screen as you would to refresh Mail. Once it finishes the refresh, click Done and I think it will have updated. I tried it once out of curiosity and it worked. Sample size of one...