I've had/seen several discussions were someone has made a statement that, in their mind, absolutely cements their argument, but instead leaves me in stunned awe of their misunderstanding, ignorance or just plain stupidity.
I typical discussion may go like so (exaggerating for demonstrative purposes):
Me: He's 13 and shouldn't be drinking alcohol.
Other: But drinking alcohol makes him happy. You wouldn't want him to be unhappy, would you? *gives knowing look*
Me: *incredulously stunned silence*
In this example, the other person's statement demonstrates the other person completely missing the point, but acts like they've just presented a winning argument.
Another exaggerated example:
Me: I can't stand people who think heavy metal is evil.
Other: That's kind of like how I can't stand redheads.
Me: But that's genetic!
Other: Now you know how stupid you sound.
In this example, the other person's argument may sound perfectly reasonable to them, but to any sane person, they look like an idiot.
Is there a word that describes such a statement? I can only think of it being unintentionally ironic, as it has the opposite effect to its intention. Although, the word "fallacy" has just come to mind.