It would probably enforce the idea of The Great Filter[1]. It would also show that it's not uncommon for civilizations to develop similarly to humans. I would argue that the concept of war is a very human concept and doesn't necessarily have to apply to alien live which developed completely independently of Earth life.

1. https://en.m.wikipedia.org/wiki/Great_Filter

I think it would have less long-term impacts than many would like to thin. Human nature isn't so easily changed, and over time people who don't want to believe in the discovery simply won't. We still have flat-earthers and moon hoaxers, this needn't be any different. Even assuming that people believe it, when faced with the reality of the hard choices to be made in changing our course, how long would it take to rationalize the fear away?

Not long, I suspect, except for the usual people who would have been reasonably concerned before any such discovery.