I have an upstairs bathroom where the shower diverter seemingly went bad a couple weeks ago. It's a push button diverter, separate from the spout, located underneath the knob used for turning on the water. Initially water would shoot from the diverter button. I was able to kind of fiddle around with the button and get the water to stop shooting out and the shower would still work. Not the spout though. So now when you turn the water on it shoots directly from the shower head, which is fine with me. But I'm curious as to if this will cause leaks behind the wall and unto my middle floor roof?

Initially water would shoot from the diverter button.

Yes if you water come out the diverter and have not done any repair it is most definitely leaking.