JavaFX: Eliminación item de un ListView

Main.java

import javafx.application.Application;
import javafx.collections.FXCollections;
import javafx.collections.ObservableList;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.geometry.Insets;
import javafx.scene.Scene;
import javafx.scene.control.Button;
import javafx.scene.control.ListView;
import javafx.scene.layout.VBox;
import javafx.stage.Stage;


public class Main extends Application {

    public static void main(String[] args) {
        launch(args);
    }

    @Override
    public void start(Stage primaryStage) throws Exception {

        ListView<String> listView = new ListView<>();
        ObservableList<String> items = FXCollections.observableArrayList("Uno", "Dos", "Tres", "Cuatro");
        listView.setItems(items);


        Button eliminarButton = new Button("Eliminar");
        eliminarButton.setOnAction(new EventHandler<ActionEvent>() {
            @Override
            public void handle(ActionEvent event) {
                int posicionAEliminar = listView.getSelectionModel().getSelectedIndex();
                System.out.println(">" + posicionAEliminar);
                if (posicionAEliminar >= 0) {
                    items.remove(posicionAEliminar);
                } else {
                    System.out.println("No se puede eliminar, nada seleccionado.");
                }
            }
        });

        VBox vBox = new VBox();
        vBox.getChildren().add(listView);
        vBox.getChildren().add(eliminarButton);
        vBox.setSpacing(20);
        vBox.setPadding(new Insets(20, 20, 20, 20));


        Scene scene = new Scene(vBox, 400, 400);
        primaryStage.setScene(scene);
        primaryStage.show();
    }
}