Usually, the JavaFX application closes once all of its windows (Stages) are closed.
You can close the application at any moment by calling javafx.application.Platform.exit().
Don't call System.exit() as you may be used to doing in Java programs. By doing that you break the JavaFX application workflow and may not call important logic written in on close handlers such as Application.stop().
If you don't want your application to automatically close, add the following code at the beginning of your program:
javafx.application.Platform.setImplicitExit(false);