Is the theory of incidence geometry complete?

As Greg explains, the theory of projective planes obeying Desargues is basically equivalent to the theory of division rings, while the theory of projective planes obeying Desargues and Pappus as equivalent to the theory of fields. I haven't seen an axiomitization of projective planes with betweenness, but I assume that this would turn into the theory of ordered fields.

To finish the answer, one should say that none of these theories are complete. For example, the Fano plane is realizable in $KP^2$ if $K$ has characteristic $2$, but not otherwise. There is an example, which I am too lazy to draw, of an arrangement of points and lines which is true in $KP^2$ if and only if $K$ contains a square root of $5$. Thus, $\mathbb{R}P^2$ can be distinguished from $\mathbb{Q}P^2$, even though both obey Desargues and Pappus, and presumably whatever axioms of betweenness you want to impose.

You should be able to adapt the proof of Mnev's universality theorem (see also) in order to show that, if $K$ and $L$ are fields which can be distinguished by some first order property, then the projective planes $KP^2$ and $LP^2$ can be similarly distinguished.

Pappus theorem implies Desargues'. The theory is far from being complete, not in the logical but in the philosophical and even aesthetic sense. Why do these incidence identities look so beautiful? :) Also what about combinatorics of, say, free or projective or some other submodules of the free module over the ring? noncommutative? (There was some old activity on this subject.) What about combinatorics of geodesic surfaces in nice Riemannian manifolds, and so on?

Wikipedia says that it is a theorem of Hilbert that any projective plane that satisfies Desargues' theorem is the projective plane which is the set of lines thruogh the origin in $D^3$, where $D$ is a division ring. As Wikipedia also explains, you also need Pappus' theorem to know that $D$ is commutative. Then you are more or less done.