  The Fredholm property of Toeplitz operators on the $p$-Fock spaces
$F_\alpha^p$ on $\mathbb{C}^n$ is studied. A general Fredholm criterion for
arbitrary operators from the Toeplitz algebra $\mathcal{T}_{p,\alpha}$ on
$F_\alpha^p$ in terms of the invertibility of limit operators is derived. This
paper is based on previous work, which establishes corresponding results on the
unit balls $\mathbb{B}^n$.
