iPhone: Interaction between applications and mobile Safari is possible, as demonstrated by Alocola. Applications register to handle links, and can also issue GET requests through the browser.
Android: It's possible to register an intent-handler within your application, though it it appears to only be relevant for non-webkit based interactions. For whatever reason, clicking custom protocol links fall flat.
WebOS: It doesn't appear to be possible. There's a resource file that maps internal applications to protocols, but it appears to be a reference and not modifiable.