$G$-functions are power series in $\Qbar[[z]]$ solutions of linear differential equations, and whose Taylor coefficients satisfy certain (non)-archimedean growth conditions. In 1929, Siegel proved that every generalized hypergeometric series ${}_{q+1}F_q$ with rational parameters are $G$-functions, but rationality of parameters is in fact not necessary for an hypergeometric series to be a $G$-function. In 1981, Galochkin found necessary and sufficient conditions on the parameters of a ${}_{q+1}F_q$ series to be a non polynomial $G$-function. His proof used specific tools in algebraic number theory to estimate the growth of the denominators of the Taylor coefficients of hypergeometric series with algebraic parameters. In this paper, we give a different proof using methods from the theory of arithmetic differential equations, in particular the Andr\'e-Chudnovsky-Katz Theorem on the structure of the non-zero minimal differential equation satisfied by any given $G$-function, which is Fuchsian with rational exponents.