Unfortunately the GameView class is an internal class so the only way would be to use reflection to make your way down to the actual GameView window.

Here's a similar case on how to access internal classes via reflection:

http://answers.unity3d.com/questions/129694/can-you-turn-off-auto-keyframe-in-the-animation-wi.html

But be careful! Those classes are internal and they can change them at any time. It's something you actually shouldn't use if you can avoid it.

edit

I've written a little helper function which will return the size of the main GameView:

// C#
public static Vector2 GetMainGameViewSize()
{
    System.Type T = System.Type.GetType("UnityEditor.GameView,UnityEditor");
    System.Reflection.MethodInfo GetSizeOfMainGameView = T.GetMethod("GetSizeOfMainGameView",System.Reflection.BindingFlags.NonPublic | System.Reflection.BindingFlags.Static);
    System.Object Res = GetSizeOfMainGameView.Invoke(null,null);
    return (Vector2)Res;
}

using UnityEngine;
using UnityEditor;

public class AShimSetGameSizeWindow: EditorWindow {

    private Vector2 _size = new Vector2(800, 600);

    [MenuItem ("Window/Set Game Size...")]
    static void Init () {
       AShimSetGameSizeWindow window = (AShimSetGameSizeWindow)(EditorWindow.GetWindow(typeof(AShimSetGameSizeWindow)));
    } // Init()

    public static EditorWindow GetMainGameView() {
       System.Type T = System.Type.GetType("UnityEditor.GameView,UnityEditor");
       System.Reflection.MethodInfo GetMainGameView = T.GetMethod("GetMainGameView",System.Reflection.BindingFlags.NonPublic | System.Reflection.BindingFlags.Static);
       System.Object Res = GetMainGameView.Invoke(null,null);
       return (EditorWindow)Res;
    } // GetMainGameView()

    void OnGUI () {
       _size.x = EditorGUILayout.IntField("X", (int)_size.x);
       _size.y = EditorGUILayout.IntField("Y", (int)_size.y);
       if (GUILayout.Button("Set")) {
         EditorWindow gameView = GetMainGameView();
         Rect pos = gameView.position;
         pos.y = pos.y - 0;
         pos.width = _size.x;
         pos.height = _size.y + 17;
         gameView.position = pos;
       }
    } // OnGUI()

} // class AShimSetGameSizeWindow