Yes, but… I can’t help but put myself in the shoes of your friend. Who may not know what you’re actually MEANING when you say, “I’m fat.” Because all the other things you say are true — most of us do equate “fat” with “unlovable,” “worthless,” etc. And if that did happen to be what was running through your head when you said, “I’m fat,” then to have your friend respond, “Yes, you are; how do you feel about that?” would probably make you feel very, very bad. And friends basically don’t want you to feel bad.
It’s not that I don’t agree with your basic premise — I do. But I think you’re being pretty hard on your friend, who probably had no idea what to say in response that wouldn’t offend you. Just a thought.