As mentioned by Will Sawin, a necessary condition is that $p$ divides $n$. Thus let us assume that $n=pk$. Denoting $e_1,\ldots,e_n$ the canonical basis, the knowledge of the $p\times p$ minors is the knowledge of the $p$-vectors $$(Ae_{i_1})\wedge\cdots\wedge(Ae_{i_p})\in\Lambda^p(K),$$
where $K$ is the field of scalars (e.g. $\mathbb C$).

Splitting
$$(Ae_1)\wedge\cdots\wedge (Ae_n) = [(Ae_1)\wedge\cdots\wedge (Ae_p)]\wedge\cdots\wedge[(Ae_{n-p+1})\wedge\cdots\wedge (Ae_n)],
$$
we see that $(Ae_1)\wedge\cdots\wedge (Ae_n)$ is a polynomial function in the $p\times p$ minors. Since $(Ae_1)\wedge\cdots\wedge (Ae_n)=(\det A)e_1\wedge\cdots\wedge e_n$, we deduce the value of $\det A$.