Is it OK to use "empty-handed" on an animal?
Solution 1:
It is most definitely acceptable to use the phrase empty-handed to describe animals.
Even if the animal you're referring to has no hands, the phrase empty-handed is understood to mean "having acquired nothing."
Solution 2:
You can write simply came up empty to indicate that its attempt was unsuccessful. (see m-w.com's definition, below item 6.)
As pointed out in comments, "dropped itself into the water" sounds a bit off. I would suggest something more like this:
One of the seagulls spotted a fish and dove after it, but came up empty.