For $d \ge 2$, let $X$ be a random vector having a Bingham distribution on $\mathcal{S}^{d-1}$, the unit sphere centered at the origin in $\R^d$, and let $Σ$ denote the symmetric matrix parameter of the distribution. Let $Ψ(Σ)$ be the normalizing constant of the distribution and let $\nabla Ψ_d(Σ)$ be the matrix of first-order partial derivatives of $Ψ(Σ)$ with respect to the entries of $Σ$. We derive complete asymptotic expansions for $Ψ(Σ)$ and $\nabla Ψ_d(Σ)$, as $d \to \infty$; these expansions are obtained subject to the growth condition that $\|Σ\|$, the Frobenius norm of $Σ$, satisfies $\|Σ\| \le γ_0 d^{r/2}$ for all $d$, where $γ_0 > 0$ and $r \in [0,1)$. Consequently, we obtain for the covariance matrix of $X$ an asymptotic expansion up to terms of arbitrary degree in $Σ$. Using a range of values of $d$ that have appeared in a variety of applications of high-dimensional spherical data analysis we tabulate the bounds on the remainder terms in the expansions of $Ψ(Σ)$ and $\nabla Ψ_d(Σ)$ and we demonstrate the rapid convergence of the bounds to zero as $r$ decreases.