Inasmuch as Albert Michelson grew up around Virginia City, someone must have thought that getting the Cartwrights involved in his life would make an interesting story. * It might have, but this isn't it.
Given Michelson's great intelligence, the Cartwrights try to get him appointed to the Naval Academy, so he can receive the education he deserves. But... Michelson is thwarted at every turn by his teacher (William Schallert) who doesn't like people with dark skin, or Jews, or fill in the name of any discriminated-against minority.
Of course, justice wins out in the end, and Schallert is even allowed to keep his job after promising to never, ever do bigoted things again. (Spare me.)
AS A LIBERAL, I detest this rewriting of history to "prove" that some people (eg, the Cartwrights) were good and decent and unprejudiced and just and (fill in whatever positive attributes you care to). Such stories are little more than an excuse for the viewer /not/ to feel guilty about their own failings.
"In the real world", the white citizens of Virginia City would have had little social tolerance for people who looked or acted differently. They wouldn't have given a damn whether Michelson was admitted to Annapolis. Indeed, I suspect Michelson was nominated and appointed without any problems.
* Given Michelson's stated age of 16, the story would have had to take place in 1868 or 1869. I couldn't find a chronology for the Ponderosa or the Cartwrights, so I don't know if this is accurate.