The end of the earth's life and our life as we know it may very well happen whether we explore space or we do not. What space exploration might let us do is have a place to go if this should loom immanent. If it does, another type of colony might prove a haven.
But I do not think society will experience armageddon in the same way most people do. Most think we will get blasted by a meteor or that a 3rd world war is going to break out. If armegeddon of any sort breaks out, it will be because of global climate change.
There will be an end of the world. There will be a meteor or some outside influence that completely alters the course of this planet, the people on it, and the universe that it thrives in. But until that day comes, all we can do is explore the answers in space.
The odds are society will experience Armageddon at some point. The question is when. With all of the comets and asteroids careening through space at any given moment, it stands to reason that sooner or later we will be in the direct path of something big and fast that will not burn up as it enters our atmosphere. We need to explore in hopes of finding a safe haven.
I don't believe we're headed for Armageddon if we choose not to further explore space. In fact, I think the opposite could be true. There is a lot of space debris being generated that circulates the Earth. We have no way of knowing how dangerous this space debris could become over time; perhaps one day it could fall back to Earth. Additionally, Armageddon is something not connected to space exploration. It has to do with Biblical end times.