None of the following expected approaches seem able to show a URL in a browser when the JavaFX app has been dragged-to-install. Specifically, the problem is observed if the following steps are used to launch:
Show embedded in a browser, drag out, close the browser/tab, close the app, launch the app from the desktop link.

(a)
appletStageExtension.showDocument(url);

(b)
def applet = FX.getArgument("javafx.applet") as java.applet.Applet;
def appletContext = applet.getAppletContext();
appletContext.showDocument(url, "_blank");

(c)
def basicService = ServiceManager.lookup("javax.jnlp.BasicService") as BasicService;
basicService.showDocument(url);

When run as an Applet or as a WebStart app, these work as expected.

Produced on JavaFX 1.2 + Java 1.6.0_13 on Linux and Windows XP. (not yet tried on Mac)