Some background info from a Navy helicopter maintainer
The USS Chancellorsville is a small boy. Helos attached to small boys primary mission is usually vertical replenishment, search and rescue, and or antisubmarine warfare. For these type of missions the birds are not usually outfitted with weapons larger then a 9mm. For antisubmarine warfare the birds can be outfitted with torpedoes or missiles.
If the Navy had fired back they could have easily used the USS Chancellorsville weapons and made the pirates ship into a new coral reef and not endanger the crew of the helicopter.
I have no opinion either way about not firing back. Not in the mood for politics today.