Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It is also the only one to be a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In [this paper][1] Kostant and Sternberg work out the conditions under which $G.[v]$ is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's *Symplectic Techniques in Physics*.

The general case is thoroughly studied in J. A. Wolf, [The action of a real semisimple group on a complex flag manifold][2]. 


  [1]: http://www.ams.org/mathscinet-getitem?mr=661285
  [2]: http://www.ams.org/mathscinet-getitem?mr=251246