It is unknown what this war was fought over, or how long it lasted.
It's possible that the Moon's traditional blood sport Punishment Soccer is related to this war, however, as the Moon's rebellion was over 76 years ago, it's extremely unlikely that this event is referring to the original rebellion.
In an alternate timeline described in Crisis on Infinite Marses part 2, the War for Earth's Moon was a much bigger conflict, and the lawmen from Mars were killed, leaving Mars a much more lawless place. As a result, nearly all human settlers were killed by robot outlaws, along with some, or all martians. The exception was The Red Plains Rider, who grew up tough and with a thirst for vengeance against all robots.