Minutemen aren't one of the main factions per se. It's only the BOS, Institute, or Railroad. To get the final judgement, you would have to side with either the institute or railroad.
*edit* Decided to take a look at the wiki, and found: "If you are currently siding with the Minutemen and you want to obtain this weapon without siding with the Brotherhood of Steel, just continue the Railroad quests until you receive the quest http://fallout.wikia.com/wiki/Precipice_of_War, but do not go to Police Station. Instead go to The Prydwen, and kill Elder Maxson in order to receive Final Judgment.". So yeah, you're basically siding with Railroad.