I want to use a remote URL (like a favicon) for a notification icon, however the constructor for a notification accepts only a resource ID (which is an integer) instead of s
You can't do this in Android currently, because resources can only be loaded from the /res/ directory of your APK. See Resources and Internationalization. The /res/ directory is built when you create your APK and can't be changed after.
Do you expect your Icon to change often enough that you need to query a URL to ensure it is up to date every time you send a notification?
Most applications that need to change their icon would simply put an update out on the Android market.