The stakes are no less significant for Columbia, Harvard and other private universities that face similar rumblings among their graduate students. To Columbia, these students come to the campus to earn degrees, and should not be viewed as university employees.

“We respect the rights of students to express their views and the rights of all students and faculty to continue their teaching, learning and progress toward degrees,” said Caroline Adelman, a Columbia spokeswoman. But, she added, “we do not understand why the G.W.C.-U.A.W. prefers the pressure tactics and disruption of a strike to a definitive, nonpartisan resolution of that legal question in the federal courts.”

The question of whether graduate students at private universities can unionize has seesawed for years, depending, in part, on the political leanings of the members of the National Labor Relations Board. During the Clinton administration, the N.L.R.B. established the right to unionize in a case that involved graduate students from New York University. A few years later, though, that decision was reversed during George W. Bush’s presidency with a case out of Brown University. Then, during the Obama years, in 2016, the N.L.R.B. ruled again, this time in favor of the Columbia graduate students.

But Columbia has declined to bargain, and with a Trump administration that is less friendly to unions now in charge, the case is expected to eventually find its way before a federal appeals court.

If the court rules in Columbia’s favor, the decision could have repercussions for graduate students at other private universities. The U.A.W. represents graduate students at the New School and N.Y.U., both private schools that voluntarily recognized their unions, as well as at Boston College and several public universities. Other unions, like the American Federation of Teachers, are also active in the movement to organize graduate students.