First, let me clarify some distinctions; a Marine is certainly not a Soldier. Soldier is Army, Airman is Air Force, and a Sailor is Navy.
Second, the story is false. As someone with an 'inside scoop', the President wouldn't (and cannot) 'relieve' an enlisted military personnel on the spot.
The closest I could find to this story is this article.
A 26-year-old sergeant and Iraq veteran is in the process of being
dismissed from the Marine Corps after making "political statements"
about the commander-in-chief on Facebook.
Such comments are considered detrimental to "good order and
discipline" under the Uniform Code of Military Justice.
It is just that, enlisted personnel are under a contract with the government to serve the time period they signed for unless violating a Punitive Article of the Uniform Code of Military Justice, medical separation, or a few other special circumstances. And for every circumstance, there is a board of high-ranking, competent personnel who make the decision to separate an individual. If the President had in fact said such a thing, it would be in direct violation of the Marine ethos of being able to correct anyone, regardless of rank (In the correct manner, of course).
I'm going to chalk this up to President-bashing, which has been gaining increased momentum over the past decade. The likelihood that the President of the United States (who is ALWAYS shown in the world's eye) would wear the American Flag pin upside-down, which is disrespectful of our country in almost every way, is low. He has teams of people to look over his appearance before stepping out and into the public's view.
If anything like his had happened, it would be all over the military press (Yes, we have a press).
EDIT: I have a few more references and points to make (For quality assurance, of course);
First, on the unverified Wikipedia page:
Below are some of the most common reasons for discharge:
- Expiration of Term of Service (ETS)
- Reaching the maximum age limit
- High Year Tenure (reaching the maximum allowable time-in-grade, and not selected for promotion)
- Disability, Dependency, or Hardship
- Pregnancy/Parenthood
- Personality Disorder
- Condition not a disability
- Physical or Mental Conditions that interfere with military service resulting in being placed on the Temporary or Permanent
- Disability Retirement Lists
- Convenience of The Government/Secretarial Authority (voluntary redundancy due to funding cutbacks, for example)
- Unsuitability
- Misconduct - Minor Disciplinary Infractions
- Misconduct - Drug Abuse with and without administrative review board
- Misconduct - Commission of a serious offense
- Entry-Level Performance and Conduct
- Resignation (available to officers only)
- Reduction in Force (RIF)
- UnCharacterized if discharged within the first 180 days of service and no misconduct found in service members record
- Punitive Discharges - Bad Conduct Discharge issued by either Special Court Martial or General Court Martial only. Dishonorable
- Discharge issued by a General Court Martial only
This all seems very vague, though what isn't covered here is likely covered underneath an Administrative Separation Board for review.
Second, this page goes over the process pretty extensively, specifically
Once the commander receives the attachments from the respondent,
he/she determines whether or not to proceed with the discharge
proceedings. If the commander elects to proceed an administrative
discharge board is convened (if required). If the board is not
required, the commander forwards the package to the approval authority
(usually the installation commander) for final approval or
disapproval.
The President would certainly have to go through the proper military channels in order to separate a service member.
