I just did it and it works great. Here's the helpful response I got from the man who invented the app I needed to download on my knidle fire.
thank you very much for your request. Unfortunately does Amazon not allow SmoothSync for Cloud Calendars to be installed on the Kindle Fire. Technically it would run without issues.
However, you may want to try our app "CalDAV-Sync". It is basically the same as SmoothSync for Cloud Calendars but you would have to explicitly set the server to "iCloud" as it also supports other servers.
Now, don't ask me why Amazon allowed CalDAV-Sync but not SmoothSync - I have absolutely no clue.
As he says, just remember to use iCloud as your server name on the app.
You'll love it.