The king got away with it because he's a badass. Tenchi, a no-account nobody from a backwater planet (in the eyes of the people, at least) would have a helluva time... though the fewer, the better. Most likely, if it were to happen, it'd only be Ryoko and Aeka.
Nevermind the fact that due to the fact that Washuu is Tsunami's sister and Mihoshi's great trandmother (Or ancestor of some sort, at least) that effectively makes any path Tenchi takes incest. Whee!
I wish he'd go with Ryoko myself, but I've heard that he's chosen Sasami in the novels. I'm not entirely sure which novel so I can't tell you if it's Kajishima canon or not.