You can authorize your Soldiers to go the gym as long as it does not violate any orders you have been given by your superiors. If you allow a Soldier to go to the gym because your mission is complete, that is your call.
It is important to understand that you are responsible for your decisions. If a superior wants to know where someone is and you tell them you sent them to the gym, be prepared for their response. It could be okay or it could be a significant emotional event. Either way, you made the call, you keep the responsibility. If you want to be proactive (and I suggest you do), contact your supervisor prior to making your decision on the gym trip.
You should also be prepared to tie up loose ends within your squad. If you have other Soldiers who do not want to go to the gym, what will you tell them when they ask to go downtown for a few hours or close up shop early and head home for the night? How will you handle their request?
My recommendation would be to have hip-pocket training scheduled and use this time to train Soldiers on skills they need practice on. With that stated there may be a time when it would be appropriate to give a Soldier some downtime as a reward for a job-well-done. Just make sure your Soldiers understand it is important to stay out of trouble. Soldiers with time on their hands have been known to make poor decisions.