A mid (aka a person on active duty in the Navy) won't necessarily get in trouble for seeking care without prior authorization from Tricare; however, it can possibly lead to all sorts of issues.
The first call for any active duty person when they think they need care outside of their Tricare Region should be to Tricare. (Obviously this does not apply to Emergency care covering life, limb, or eyesight). The reason for this is that there may be a military treatment facility nearby that can treat you and will therefore cost the military significantly less money.
If you seek care outside the military system without prior authorization you could be billed for the full amount owed and Tricare could deny coverage. Now, in practical application I have personally never heard of this occurring, but just as with other insurance programs Tricare works just the same. They are looking to recoup as much money as they can.
As an aside: I hope your mid used their military ID card and their Naval Academy address when checking in. That will save you, the parents, many a headache come billing time.