Nah, I think that's right. It does imply a contract.
I think the key here though is that they are not suing for a violation of Constitutional rights, so state action is irrelevant. They are suing for a violation of the Civil Rights Act of 1866. Even if they win, Congress can change that. I wonder if Congress would, though.
There is also the question of whether a Congressional enactment can regulate purely intrastate commerce. I'm not sure that anyone has raised that, though.
Under the ever broadening definitions, it could fall under Interstate Commerce...
Some of the food in the cafeteria is from Mainland sources?
Some of the faculty, staff, and administration are hired from out of state?
Their books and supplies, including the library books, are mostly obtained, directly or indirectly, via Interstate Commerce, and are an integral portion of the institution's operations?
The money they use to pay salaries and bills; and that they accept for tuition, comes from mints on the Mainland, and/or substantially from sources engaged in Interstate Commerce?
And that money passes through banks engaged in Interstate Commerce?
Isn't EVERYTHING Congress wants to regulate somehow construed as "Interstate Commerce"?