What does "I like girls just fine" mean?

It means the same thing as "I like girls." The connotation is slightly different and the reason for including "just fine" is to counter the accusation that Nathan does not like girls.

An example in a different area:

Would you like some tea?

Tea would be just fine. / I like tea just fine.

The expression is used as a basic or stoic acceptance or acknowledgement. Other words with similar usage is "okay" or "alright". Each carry their own typical uses and connotations but the gist is there. A wordier way to say it:

I like girls as much as the next guy.

Notes: I was interrupted while answering so there are more than the typical amount of edits. I apologize for any confusion.

I believe in this context it means "sufficiently", with the definition of sufficiency actually being defined by the person who asked the original question. In other words, the respondent is stating that he meets the questioner's expectation of liking girls.

Sadly, I can't think of another example at present. Sorry.

EDIT: Based on the conversation below, I revise my answer somewhat -- either conversant's views may be in use. However, the core point is still the same: it means "sufficiently".

Examples...

"We've run out of beer! I'll just pop down the shop and get some"

"Don't bother, wine will be just fine" (implication: good enough for me)

...and...

You don't really like me, do you?

Rubbish! I like you just fine (implication: enough that you should be satisfied)

Examples seem to be in high demand...

SUE: Would you like some mustard?

JOHN: Mustard would be just fine.

What the speaker is saying is that he does, in fact, like girls. Just as John (in my made-up example) likes mustard.

However, with a certain emphasis, the speaker could be implying that he likes girls, but isn't attracted to them.