This is part of the Bore-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 exposition. As he notes there, the algebraicity you ask about is a consequence of Chow's theorem. Also, the orbit of the highest weight space under the compact real form $G_0$ is open by dimension and closed by compactness, so it coincides with the (algebraic) $G$-orbit.

This is part of the Borel-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 exposition. As he notes there, the algebraicity you ask about is a consequence of Chow's theorem. Also, the orbit of the highest weight space under the compact real form $G_0$ is open in the $G$-orbit by dimension and closed by compactness, so it coincides with the $G$-orbit.