I'm a Federation Bounty Hunter, at the rank of Admiral with a cruiser, tons of missions, a fortune, and a 90% approval rating from the governor of the Sol System.

I've read that if the governor likes you enough, you get a chance to marry his daughter. Well, 90% seems pretty good to me. So far, however, I haven't had a chance to do that.

How does the whole marriage thing work in this game? Is there a certain event I need to trigger beyond whatever approval rating I have?

Actually you only need 75%+, but you have to talk to him often. Afaik there's no explicit trigger, you'll randomly get the chance to meet his daughter and flirt with her. If she falls in love with you, you can ask to marry her.