Let $G$ denote a connected reductive algebraic group over an algebraically closed field $k$ and let $X$ denote a projective $G\times G$-equivariant embedding of $G$. The large Schubert varieties in $X$ are the closures of the double cosets $B g B$, where $B$ denotes a Borel subgroup of $G$, and $g \in G$. We prove that these varieties are globally $F$-regular in positive characteristic, resp. of globally $F$-regular type in characteristic $0$. As a consequence, the large Schubert varieties are normal and Cohen-Macaulay.