Can I legally omit the fact I hold a Ph.D. degree?
No. You cannot hide such a fact in US admissions processes. Lying about your previous academic record—whether by omission or by claiming false credentials—is sufficient reason to revoke an offer of admission if it is caught before the applicant matriculates, or grounds to expel a student if caught after enrollment. Depending on the magnitude of the fraud, you may even be asked to pay restitution for the funds spent.

As far as if it’s illegal, that’s a trickier issue. It basically comes down to money: if you received funding based on a misrepresentation or fraudulent application, then it’s far more likely that you’ll face civil or criminal penalties.