Dock

Ok, let's try this: Remove FaceTime from the startup if it is setup that way, and then remove the icon from the Dock.

1. Shut down and restart your Mac.
2. If the FaceTime icon still appears on the Dock, then Secondary Click the icon on the Dock (tap on it with two fingers once).
3. This will bring up a popup. Select Options. Make sure there is no tick mark against Open at Login. Also, in the same popup, select Remove from Dock

Has this solved your issue?

What happens when you try to remove it?

Make sure you've quit FaceTime app before trying...