First of all, you can't bring any US standard or rule as is to different other country such as Japan, Vietnam, China, Iraq, Cuba etc.
We are always stranger once you step out from mother country and have respect their standard if you wish to be respected by them. USA is just a country of the world and the citizen doesn't be allowed to do what they like to do outside of the country.
Talking about blood based discrimination in Japan, you have to consider well their background and culture because Japan has not been Melting Pot like USA where immigrants have invaded natives.
It like "no foreigners" sign means their instinct for fear to stranger historically. They don't care what skin color do you have. If you speak Japanese fluently and show them no intention of malice and/or disadvantage, you are welcomed anywhere anytime even at Onsen or something else ever.
DNA thing mixed to this kind of discrimination is totally silly view point Western people often tend to self mislead.
If stranger can understand Japanese and their culture with respect, he must follow their local standard so that he'll be accepted, that's all.
If he can't speak Japanese, he'd better not come close to any place "no foreigners" for avoiding any possible unpleasant result each others. It is a general etiquette for traveling abroad.
We can't force our own standard onto different people because it is almost kind of insulting for them.
If he really wants to bathe at Onsen, make his own Onsen so that he will be able to do anything he like in USA.