One of the most annoying things about race in America is that it takes complex notions of culture, nation, and identity and tries to map them onto the mind-numbingly simple binary of black and white. Or more accurately, white and non-white.
The justices never said what whiteness was, only what it wasn't. These days, the term minority has come to be a more inclusive version of blackness and the levels of racism are far less intense than ever before but much of the same racial dynamics still apply. For example, consider that there is now PC term for whiteness that corresponds with minority. In theory, the term should be "majorities", but have you ever heard of anyone refer to someone as a majority or majorities? Minority status and what it means to be a minority in the US is a topic that people spend a lot of time trying to describe and understand but "whiteness" doesn't even have a similar term and is a fitting topic only for race conscious comedians.
Like in the Ozawa case and the Thind case, does America construct a common notion of race by defining what whiteness is not and not what it is? Although far from universally accepted among all American subcultures, my gut tells me that in the mainstream there can be no common definition of what whiteness is because to a large degree to be white is to be human. Whiteness is meaningless* because a white person is just a normal human or a member of what could be called the normative race while are normal humans plus their racial identity.
I would call this an assimilation based definition of race because at it's heart it constructs the idea of race as a measure of assimilation into mainstream American culture. Whiteness means full assimilation and normative status. Minority status means having an additional non-assimilated identity. There is the sense that if you're white you would have to add on blackness or asianness to become more black or asian but if you're black or asian you would need to remove your blackness or asianness to become more white.
It is important to note that unlike in the past this additional status is no longer viewed in solely negative terms and is even described as desirable.** We've certainly come a long way since the days of the California Alien Land Law (which by the way wasn't struck down until 1952 in Sei Fuji v. State of California). But on the other hand, I do have a problem with this view of race as favor or race as bonus identity because it ends up trivializing*** the severe racial biases in current society and tends to overlook the extremely dark history of American race relations. These biases are so easy to find in the US legal system, the health care system, the education system, the political system, and the economic system that they pop up with even a cursory analysis.
The blog Stuff White People Do is an outstanding blog that takes a deep look at whiteness and what it means in the context of non-white identity. Highly recommended.
*Here is a great example of a meaningless description of whitness.
**For an example of a blog absolutely dripping with the notion that minority status can become desirable check out this blog.
***It doesn't get much more trivial than this.