Native Russian speaker here: translation „military“ is correct, because this is how it is positioned in official Russian sources - an operation conducted by military forces (the term operation comes from FSB, meaning that according to Russian propaganda it is more like police or counter-terrorism operation).

Slavic speaker here, “Военная” looks like the adjective derived from the “war” noun. So not “war”, but kind of like, “war-ful operation”, or “combat operation”.

It is indeed derived from „war“, but here it has different meaning, so „military“ is correct translation. You would use „war operation“ as translation if this operation was an episode of war, but according to Russian propaganda it is not. Combat operation would be translated as «боевая операция».