Demonstrates adding a basic alert handler to a JavaFX WebView.

WebViewWithPromptHandler.java

import javafx.application.Application;
import static javafx.application.Application.launch;
import javafx.event.EventHandler;
import javafx.scene.Scene;
import javafx.scene.control.Label;
import javafx.scene.layout.StackPane;
import javafx.scene.web.*;
import javafx.stage.*;

public class WebViewWithPromptHandler extends Application {
  public static void main(String[] args) { launch(args); }
  @Override public void start(final Stage primaryStage) {
    WebView webView = new WebView();
    webView.getEngine().loadContent("<button onclick='alert(\"Alerted\"); alert(\"Alerted 2\");'>Popup</button>");
    webView.getEngine().setOnAlert(new EventHandler<WebEvent<String>>() {
      @Override public void handle(WebEvent<String> event) {
        Stage popup = new Stage();
        popup.initOwner(primaryStage);
        popup.initStyle(StageStyle.UTILITY);
        popup.initModality(Modality.WINDOW_MODAL);
        
        StackPane content = new StackPane();
        content.getChildren().setAll(
          new Label(event.getData())
        );
        content.setPrefSize(200, 100);
        
        popup.setScene(new Scene(content));
        popup.showAndWait();
      }
    });

    final Scene scene = new Scene(webView);
    primaryStage.setScene(scene);
    primaryStage.show();
  }
}

Answer to JavaFX forum question: On alert handlers and WebView

For nicer popup dialogs, the ControlsFX dialog classes can be used to define the popup dialogs.