Note that the issue of what steps a college may take to provide advantages to URMs in admission is different from the issue of whether a college is discriminating against Asian students in favor of white students. With respect to URMs, colleges admit they are practicing affirmative action, and the issue is about how, exactly, they are legally allowed to carry that out. With respect to Asians, colleges deny that they are discriminating, and it will be very difficult to obtain proof that will be enough to sustain a claim that they are.
The Fisher case is about affirmative action for URMs. I suspect that private colleges are interested in it because it could change what is acceptable in terms of running an affirmative action program. The case is unlikely to be significant on the question of claimed discrimination against Asians.