I don't remember having trouble navigating in iOS 7 like I am in 8, but perhaps that's because I don't use Calendar (I use Fantastical). Sorry for the confusion!
Regardless, I think we're talking at cross-purposes. I consider Calendar to have two "day views"; one with a month visible on top and one with a week visible on top. The behavior of the button to the left of the search is very different in these two views, but tapping it should do something.
If nothing happens when that button is tapped, then there's a bigger problem.
Restarting the phone didn't help, but possibly resetting will:
Turn your iOS device off and on (restart) and reset
I doubt that will fix it, as it sounds like the app is screwed up. But it's a quick thing to try.