We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $λ$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,λ)$ and $p_u(M,λ)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & Müller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument.