Survived out of desperation. Improved out of spite

So to the dad, anon just challenged his authority and his manhood. Dad is already threatened by his daughter having a boyfriend (and potentially eventually moving out due to that boyfriend/marriage), now anon is saying "I don't give a shit about you or your threat" as far as the dad is concerned. So anon saying he didn't mean it that way is what's triggering the "liar" comment.

It's a very patriarchal woman as property thing that crops up especially among white Southern Christians. I would lay money based on this story and my own experiences that anon is the girl's first boyfriend and that she's in the kind of family environment where she will live with her parents until marriage, a very literal dad giving daughter away at the altar situation.

Anon probably does not have sisters and didn't have all of this subtext made text constantly throughout his life, even though him carrying on a daily basis suggests some level of immersion in gun culture

Dad shows gun = "If you hurt her, I will kill you."

Boyfriend shows gun as well = "I also suspect you of being capable of hurting her and anticipate having to defend her."

And "hurt" in a dating context (or at least this specific gun related one) usually just means heartbreak, but "hurt" in a father/daughter context would more often be abuse.

So there's an implicit accusation he could read into it that turns the conversation into:

Dad: I think this relationship might not work out, and you will hurt my daughter's feelings.

Boyfriend: Well, I think you are physically harming your daughter, and brought this gun to show you I mean to rescue her.

Dad: How dare you! At least accuse me outright, for such a strong claim as this.

Boyfriend: I shall never say it outright, and therefore you have no evidence against me to use to kick me out right now.