  In this paper, we study capillary graphs defined on a domain $\Omega$ of a
complete Riemannian manifold $M$, where a graph is said to be capillary if it
has constant mean curvature and locally constant Dirichlet and Neumann
conditions on $\partial \Omega$. Our main result is a splitting theorem both
for $\Omega$ and for the graph function on a class of manifolds with
nonnegative Ricci curvature. As a corollary, we classify capillary graphs over
domains that are globally Lipschitz epigraphs or slabs in a product space $M =
N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci
curvature, including the case $M = \mathbb{R}^2,\mathbb{R}^3$. A technical core
of the paper is a new gradient estimate for positive CMC graphs on manifolds
with Ricci lower bounds.
