To be honest i think they have to be over 16 years to use the gym facilities but i would recommend that you contact Royal Caribbean via their contact details on their web. I have asked them questions in the past and they usually get back to you within 2 days.
Can confirm that Smith is correct, whether they may make an exception I don't know. It never hurts to ask the question however in our world of health and safety and law suits I'm sure you can imagine what their answer will be. Let us know the ourcome!!!