It may not sound as "natural" but indeed the correct* version is:
the moon is as beautiful as she.
She is a predicate nominative which is indeed in the subjective case. If you expand the sentence, it becomes clear:
the moon is as beautiful as she [is].
Alternately if you said
she is as beautiful as the moon.
It is clear.
Note that "than" another comparative preposition essentially starts a new clause too:
She is taller than I [am].
*Side Note on the word "correct" in this context:
On a philosophical level, some are questioning whether grammar is prescriptive or descriptive. There is a certain backlash against the usage of the word "correct" amongst those who like to say there are no such things as "rules" when it comes to grammar. Much like New Math there is a style of learning that says rules your English teacher taught you are meant to be broken. Grammar, they say, is strictly descriptive, and it is a poor writer who does not rise up in rebellion against them.
I was intentional in its usage, but wish to clarify. "Correct" means adhering to the "rules" and technically, this would only be possible when looking at a particular style guide or in the confines of an English class or, as in my case, having a mom who was an editor :) Put another way, this is the answer my teacher would have given, and I make no apologies for it. It is perfectly common and understandable to hear "than her" but if you are speaking with your grandma, she'll probably correct you - it is one of those "rules" that for better or worse is out there.
I'm not going to argue whether or not this "rule" exists, because is it does. I'm also not going to argue that the rule is iron clad, because frankly, it isn't. The point of the word "correct" is that amongst grammar nazis, this would be the "correct" rule.
It has a certain linguistic sense to it as well, as explained above. But understand that just because I state it is "correct" does not mean that the opposite isn't common.