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What does this really mean? Does it mean anything?Yes, a lot of things. And it's pretty trivial in its meaning really (also in its expression). In other words, it's not like some academic obscurantism. She clearly says something clear itself.
At first I thought your disagreement stemmed from unfamiliarity with European philosophy, and wanting everything to be spelt out in the dry style of anglosaxon philosophy.
But since you're OK with Lacan, Foucault, and Derrida, I can't possibly fathom where your disagreement with the above is.
If you can read Lacan, Foucault, and Derrida, you can surely parse Paglia (at least the above except). As for the vitriol she expresses against them, well they too have expressed vitriol against each other, at some point or another. It's par for the course, as these philosophers don't pretend to express some kind of scientific understanding (which converges), but ways to look at the world (which can diverge).
As for what she says, I'll try putting it more crudely below:
1) "In the beginning was nature".
Civilization and humans start from being parts of nature (e.g. we start as regular animals).
2) "The background from which and against which our ideas of god were formed, nature remains the supreme moral problem".
Our religion (and other ideas, philosophy, morals, etc) evolve from our initial animal status and in connection to nature (e.g. consider how early gods where embodiments of natural forces, but also how natural constraints, from death to drought, to emotions, to reproductive urges, etc where things we responded with religion, song, myth, etc.
What's "natural" to us (and conversely, unnatural/obscene/etc) is our most common moral question.
3) "We cannot hope to understand sex and gender until we clarify our attitude toward nature. Sex is a subset to nature. Sex is the natural in man".
Sex and gender are tied to our natural (animal) origins (male/female organs, reproductive instinct, different physiology, etc). Sex is natural to us -- but it's still a complex issue (e.g. morally, legally, culturally). To feel comfortable and open towards sex, we should finally feel comfortable with our animal nature.
Finally, she asserts that since sex is part of nature, it is the natural part in us (e.g. not our artifice or construction, but something we inherited from our animal nature, and which connects us to it).
You could easily find similar excerpts in all three of Lacan, Foucault, and Derrida.