Had a bowhunter post a similar scenario on another site; He shot a doe and it ran into the swamp. He immediately tracked the deer only to see her walking away inside the mouth of a black bear. Don't think he tagged her. Most states have allowances for predator loss, if there are predators in the area. If the CO doesn't require a tag MY ETHICS would say no.
BTW for the Noah Webster poster
in·teg·ri·ty
(
n-t
g
r
-t
)
n.
1. Steadfast adherence to a strict moral or ethical code
This would qualify. I would be adhering to a strict code: MINE!, not yours.