To block your aircraft from being tracked publicly, you will need to submit a request to the FAA. FlightAware also offers a service that can block your aircraft from being tracked in FlightAware website and manage blocked aircraft and users.
Below are more information about your options:
FlightAware Global
FlightAware offers the Global Gold or Platinum subscription, which allows you and your designated users to track the aircraft while it remains blocked from the general public. With this subscription, you have the flexibility to control who has access to the aircraft's tracking information, ensuring privacy while still enjoying the benefits of comprehensive flight tracking. There are many other features included with FlightAware Global and you can view them on the product overview video.
FAA LADD
If you simply would like to block aircraft from being tracked by general public and do not need access to your aircrafts movements and other benefits that our global subscription provides, you may submit a block request to the FAA through their Limiting Aircraft Data Displays(LADD) program. Once the request is received, the aircraft will be blocked within maybe 2 to 3 weeks on FlightAware and other flight tracking websites.
Please note, that once blocked with LADD and without a FlightAware Global account you will no longer have access to view the aircraft's movements on FlightAware.