Edit: In Eigenvalues of the dirac operator, M. Atiyah, 1984, the following results are stated and proved. Let $M$ be a $d$-dimensional closed spin manifold, and $D$ the Dirac operator associated to a Hermitian bundle $V$ with a connection over $M$. Let $\lambda_k$ be its eigenvalues, ordered by absolute value and increasing. The following results are named after their version in the above paper.
Theorem 2.
There exists a constant $C$ independent of $V$ such that
$$|\lambda_k|\leq Ck^{1/d}.$$
Theorem 1*.
If $d$ is odd, there exists a constant $C$ independent of $V$ such that every interval of length $C$ contains an eigenvalue.
Theorem 2*.
If $d$ is odd, there exists a constant $C$ independent of $V$ such that every interval of length $Ck^{1/d}$ contains at least $k$ eigenvalues.
About the case of even-dimensional manifolds, Atiyah gives an argument showing that such a constant would have to depend on $V$, but I don't think it rules out similar theorems for fixed $V$.
Out of transparency, here is my previous answer, which in the odd case is of course way worse. It seems classical, although I cannot seem to be able to find a reference with a complete proof (Atiyah writes a sentence about it in the above article), that the Dirac operator on a closed spin manifold $M$ of dimension $d$ satisfies a Weyl type law:
$$\frac{N(\lambda)}{|\lambda|^d}\to C$$
for some known constant $C>0$ ($C_d\mathrm{Vol}(M)$, for $C_d$ a constant depending only on $d$), where $N(\lambda)$ is the number of eigenvalues with absolute value less than $\lambda$ (counted with multiplicity). I claimed before that it implied that the gap between two consecutive eigenvalues cannot be bounded below. In fact, we get
$$\liminf_{k\to\infty}\frac{|\lambda_{k+1}|-|\lambda_k|}{k^{1/d-1}}<\infty$$
so for $d\geq2$ the gaps cannot be bounded below at least on one side.
However it does not rule out large gaps; it only rules out relatively large gaps in absolute value in the sense that
$$\limsup_{k\to\infty}\frac{|\lambda_{k+1}|-|\lambda_k|}{\lambda_k}
=\limsup_{k\to\infty}\frac{|\lambda_{k+1}|-|\lambda_k|}{k^{1/d}}=0.$$
Again, I don't know if there exists Dirac operators on even-dimensional manifolds with arbitrarily large eigengaps.
A few references: Weyl laws on open manifolds (S. Moroianu, 2003), claims that this law is well-known in the case of closed manifolds, and that the case of compact manifolds with boundary is treated in Spectral Asymmetry and Riemannian Geometry. I (M. F. Atiyah, V. K. Patodi and I. M. Singer, 1975). It is also possible that the result of Moroianu holds trivially in the compact case, see Theorem 3. The more recent work The Dirac spectrum on manifolds with gradient conformal vector fields (A. Moroianu and S. Moroianu, 2007) states in the introduction that
[The eigenvalues of the Dirac operator on a closed spin manifold] grow at a certain speed determined by the volume of the manifold and its dimension.
which I read as “the Weyl law holds for the Dirac operator on compact spin manifolds.” I bet someone with more experience with Dirac operators than myself will be able to find the missing pieces.
