I would highly suggest it become record-able, especially if it happened at work. Where I manage, regardless if the employees seeks medicial treatment or not, the incident should be recorded. Here are the reasons:
1. To prevent this injury again the the future for all employees, update you safety SOPS if needed. Maybe your policies missed something.
2. If they are actually injured after their few days of rest, they can have the quick option of medical treatment, and not have to wait for you to complete paperwork.
3. Injuries can get worse and the employee can deteriorate rapidity, especially after the adrenaline is gone. Have paperwork ready!
4. In case the employee files a complaint to OSHA or the state, you need to cover yourself at all cost. If the employee refuses medical attention, record it as "employee refused" have them sign it and keep it for at least 3 years. When it comes to a law suit, you need to protect yourself.

Hope this helps.

I don’t think this would be a recordable as long as a doctor has documented that the employee is fit to return to work. The employee is exercising personal choice to use vacation or PTO and not to return to work.