Many of them worry the conservative high court majority is poised to strike down or severely limit affirmative action.
"A broad ruling could have very far reaching effects and could possibly affect the pathway to opportunity for minority students at a range of universities," said Debo Adegbile, the group's special counsel. "Not just at state universities, but at all institutions of higher education."
But groups opposed to the state's policies think the time has come to rethink how affirmative action will be applied in the future, if at all.
"Using race in admissions decisions, to achieve diversity, amounts to stereotyping people by their race," said Joshua Thompson of the conservative Pacific Legal Foundation, which filed a legal brief in support of Fisher. "Racial diversity in a student body does not guarantee a diversity of experience and perspectives. It is unrealistic and wrong to try to pigeonhole people by their race."
A bigger debate
A larger social debate is whether race-conscious policies serve their ultimate purpose -- to help minority students ultimately achieve success, especially in high-profile professional positions.
"What we're seeing now is affirmative action is backfiring quite badly," said Gail Heriot, a law professor at the University of San Diego and a member of the U.S. Commission on Civil Rights.
She and two fellow Commission members filed a brief, citing recent studies that race preferences are doing more harm than good, and suggesting there are fewer African-American professionals than would have been with race-neutral methods.
"The problem is what is called mismatch," said Heriot. "As a result of affirmative action, black students in particular, Hispanic students as well, are likely to go to a school where their entering credentials put them towards the bottom of the entering class," at top schools like Harvard or Texas.
Their lower grades as a result, said Heriot, means those students are much more likely to give up on the ambition to major in science, or go into medicine or the law.
Many social scientists and civil groups, including the NAACP, reject that analysis.
"I think those studies and arguments don't really jive with the real world," said Adegbile. "The admissions officers are best situated to make the determination and some of the tradeoffs, to have a mix of students that can accomplish all of the things that college and universities are engines to do."
The high court has had an evolving record on the discretion of state officials to decide who attends their institutions.
The justices said in 2003 that state universities can narrowly tailor their admissions policies to consider an applicant's race, to achieve a "critical mass" of minority students.
But they reaffirmed existing limits -- bans on the use of quotas, extra "race" points in selective criteria, or "balancing" measures to reflect the larger population.
That landmark case from the University of Michigan is the subject of a separate pending high court case over a state ballot measure designed to eliminate any racial criteria.
A divided federal appeals court had concluded the voter-approved ban on "preferential treatment" at public colleges and universities was unconstitutional, and "alters Michigan's political structure by impermissibly burdening racial minorities."
The justices have accepted that case for review, and will hold oral arguments in the fall. It has the potential to complement, or surpass in impact, the separate Fisher case.
But her appeal is complicated by the issue of "standing" and whether Fisher should even be allowed to bring her lawsuit.
Fisher graduated from another school
She graduated last spring from Louisiana State University, where she went after being rejected by the University of Texas -- and school officials had argued she has no "live" controversy or claim necessary for the high court to intervene.
They said her only "harm" would be trying to recoup nonrefundable application fees.
Through her lawyers, Fisher said she did not want to be interviewed separately by CNN. But after October's argument she told reporters, "I hope the court rules that a student's race and ethnicity should not be considered when applying to the University of Texas."
In hearing the case, the high court made no mention of whether the standing issue would affect its ultimate ruling.
Another wildcard: Justice Elena Kagan will not participate in the case since she apparently had been briefed on the issue as the Obama administration's solicitor general, before joining the high court.