It seems to me that you are feeling unsettled because of her disturbing argument. This verb disturb is very tricky to us learners, because when we say "It is disturbing", it is disturbingly not a verb anymore; disturbing now becomes an adjective. So grammatically speaking, your girlfriend is at an advantage because interrupting cannot be used as an adjective, while both disturbing and unsettling can, as @snailplane explained. But if she said that the unsettling in "It's unsettling" is a verb, then you can say that she was wrong too. Because, in your example sentences, neither disturbing nor unsettling is a verb.
Your advantage is perhaps the face that both interrupt and unsettle are synonyms of disturb, and they are the synonyms in different senses, as @StoneyB explained. A disturbing movie could give me an unsettled feeling. However, if someone interrupted me, it would be more likely that I would feel that they were disturbing me rather than unsettling me. On the other hand, if someone keeps disturbing me, I probably will think that they are so annoying rather than unsettling. And, before I break into a conversation, I might ask "Am I interrupting (something/anything)?" first, because I know that I might disturb them by interrupting their conversation.
One last note, as I have often been reminded, each word exists for good reason. Even synonyms cannot always replace each other. And, context is the king.