Yes and no, a private organization can’t deny someone membership if a prospective member refused to make an oath to the effect of “I swear to discriminate again

Yes and no, a private organization can’t deny someone membership if a prospective member refused to make an oath to the effect of “I swear to discriminate against Asians”. So their freedom of speech ends where it infringes on the civil rights of others. Arguably, some DEI statements could come close to crossing that line.