I would clarify that to include that the orders must be lawful. If they are not, he is bound not to obey them.
That said, he made his point and he may well find himself out of the service. There was an officer who was a reservist during the Clinton years who made a critical remark about the Slickster in off duty off base civilian status and he lost his billet and his career was then over for all intents and purposes.
These kinds of officers and enlisted who go to this extent believe they are right to do so and know the stakes.
And the Army historically takes a dim view of soldiers who decide for themselves what orders are valid and what are not. There are a hundred layers between this officer and Obama, and every order in at least 99 of those layers are valid. He ignores them at his own peril.