Let $M$ be a smooth manifold. It is well known that the exterior differential $d:\Omega^r(M)\to \Omega^r(M)$ determines an elliptic complex.
Let $(P,M,\pi)$ be a principal $G$-bundle over an oriented compact riemannian manifold $(M,g)$ and $E$ an associated bundle corresponding to a linear representation (for example, one can take $E$ as the adjoint bundle $ad(P)$). Consider the exterior covariant differential $d_\omega:\Omega^r(M,E)\to \Omega^r(M,E)$ induced for a connection 1-form $\omega$.
Because $M$ carries a riemannian metric, I suspect that the exterior covariant differentials $d_\omega$ also determines an elliptic complex.
I would like to know a reference where I can found if I am right.
I assume you meant $d:\Omega^r\to\Omega^{r+1}$.
For the exterior derivative, $d^2=0$ implies we have an elliptic complex.
For a connection $A$ on a principal bundle $P$, the quantity $d_A^2$, the anti-symmetric part of the second covariant derivative, is the curvature: $$(d_A)^2\alpha = F_A\wedge \alpha$$ which is not necessarily zero.