I don't think that would be wise. The Army dispatched this case in the manner they did because the plaintiff was a non-ob officer who, while serving in the IRR, volunteered to be augmented to active duty to support an active-duty unit that was scheduled for deployment. Essentially, he volunteered for active duty so that he could file this TRO application - at least that's the way the Army viewed.
If another soldier, an active-duty enlisted man or an obligated officer, were to file a similar lawsuit, the Army would have no choice but to litigate it. It would get ugly and I really wouldn't foresee a favorable outcome for the plaintiff.
Wouldn't that mean that Obama would need to disclose his birth certificate if there is one?