First, we should remember that the government did not particularly spend any money on the air strike, except for incidental stuff like fuel costs. It already owned those missiles, which were part of a previous government spending program. Unlike profit-motive organizations that try to be streamlined, the government buys many things it never uses and doesn't necessarily replace things it does use. This is particularly true of the military. The US military has enough stockpile to use incredible force in Syria and still not need to change its future spending.
But anyway, let's talk about your question. Military is, indeed, an example of stimulus-style spending: a government program that employs people and creates demand across the economy, but which does not generate much in the way of goods and services that we would normally count as wealth. Engaging in war does increase employment and temporarily increase demand. To be clear, this should not be confused with helping the economy in the long-term. The Keynsian argument suggests stimulus spending as a way to smooth out short-term trends, but which must be paid for later. It's not a way to improve the economy overall.
The wealth of a country is measured by the goods and services it produces (or can buy). Military spending can jump start an economy full of pessimism or in a bad equilibrium, but it clearly shifts resources and people away from wealth-building and is a long-term economic drag. Think of military spending as similar to hiring people to break rocks or farm weeds, as was done in earlier periods of our history. It increases employment and but does not add to the nation's overall wealth. In fact, you can think of military spending as a transfer payment from the wealthy (who pay most of the taxes) to those who benefit from military spending (weapons manufacturers and associated industries).
As you mention, I'm not saying military spending is good, bad, immoral, or moral. Just saying economically it's a short-term stimulus, long term drag.