When the government spends a million dollars building a bomb, it also creates a bunch of jobs for the same reason.
The thing is, a bridge gets you a useful public resource that makes lost of people's lives better and encourages economic growth by making trade easier. A bomb blows up, kills some people, and disappears.
If I pay you $100 to clean my house, you've got $100, and all I've got is a clean house. Nice, but there's no market value. There's no multiplier effect.
But there is SOME utility to the gov't injecting capital in the form of military spending. As I mentioned in the comment below yours, the most extreme example of that is WWII, which finally and permanently ended the Great Depression, in a way that the WPA/etc building bridges just did not.
But remember that what did more than anything to cure this county of the great depression was the massive Keynesian capital injection that we called WWII.
Military spending is still government spending. It's going to contractors, vendors and soldiers. They are, respectively: making jobs and selling to everybody (not just the gov't), and buying cars and houses.