When is it appropriate to salute indoors?

AR 600-25 is not going to give you the answer on this. Army Regulations don't really apply to Naval Drill & Ceremonies.

You want the USMC Drill & Ceremonies Manual MCO P5060.20

1) Marines don't salute indoors.
2) We "generally" don't wear covers indoors, unless under arms (in possession of a weapon or on duty which is "figurative" armed).
3) The Marine in question is likely conforming to the Event: Cover on for appearances, but Gloves off (if we're outdoors in Blues gloves are on, if we're inside gloves are off).
4) The Anthem/Ensign rates a Salute or going to Attention, however, simply going to Attention would not "visible" on a stage, so rather than violating 1), he placed his hand on heart while at Attention which is a reasonable compromise.

So, he may have some "technical" errors but he maintains the Spirit of our (Naval Services) order in Priority and Protocol. No strict violations that I can see from the picture.

EDIT: Capt Joseph Olson Makes a GREAT point that there was "possibly" (I would say LIKELY) a Public Affairs/Information Officer there and this is a "Staged" Event, and the Marine had some "Directed" Actions.

The Location on the stage makes it "impossible" to face the Ensign (which is displayed on the screen), and "difficult" to face the Music to render Honors (without becoming a distraction to the Performer). This goes back to 3) Conforming to the Event.

During Bush 43's watch, he signed an Executive Order which provided the RIGHT, not OBLIGATION, to all US military personnel, whether active, retired, Reserves, Guard, or simply former military personnel, to salute the US flag or render a salute during playing of the National Anthem: whether indoors or outdoors, covered or not covered, in uniform or in civilian clothes. The order was intended to clear up any confusion and to spell out all possible scenarios. However, under previous directives or even generations-old, military traditions, this Marine should be saluting. That said, he may simply be performing this "hand-over-heart gesture" to emphasize to the crowd the proper manner in which we all should honor our National Flag during the playing or singing of our National Anthem. My take.

If he is wearing his cover he should salute.
From US Navy Regulations Section 1205
"Whenever the National Anthem is played, all naval service personnel not in formation shall stand at attention and face the national ensign. .... When covered, they shall come to the salute at the first note of the anthem, and shall remain at the salute until the last note of the anthem."

So the actual question is should he have been covered?

I'm guessing the ensign is on the screen behind him, so no option to face the flag and salute, although I would have turned my back to the crowd and saluted anyway. Hand over heart is a civilian thing, never did it in uniform, never should be done in uniform. During the National Anthem, you stand at attention, face the flag or music. If you're covered, then you render the hand salute.

Is it possible that this photograph was taken during the Pledge of Allegiance? Hey, I don't know. Indoors, Covered, No Gloves, No weapon. No Duty Belt. No Salute. The comments below are all solid. I know we all do things a little differently. But, not THAT differently. Lots of weird to me. Regardless, I will surmise this was a staged photo --someone wanted a Marine in Blues to add a little emphasis to the scenario.

SN Greg Wright Sgt Aaron Kennedy, MS SSgt Terry P. PO1 William "Chip" Nagel SSG Jeremy Kohlwes Capt Tom Brown

Never, if you're in the Navy or Marines. (Marines can, under arms).

PO1 Andrew Gardiner Sgt Aaron Kennedy, MS PO1 William "Chip" Nagel

Unless times have changed, when I was in, Marines only salute when wearing a cover. You do not wear a cover indoors, unless you are under arms. Now the term under arms just does not mean that you have a pistol, but you can have a duty belt and be considered under arms.

Marines should never place their hand over their heart. Since the Marine is wearing a cover, he should have saluted properly. If he is in civilian clothes or in uniform without a cover, he should stand at attention.

Great question. As I best recall when in uniform and indoors during the playing of the national anthem one typically stands at attention and doesn't render the hand salute. But then they are not usually covered. I think the rule is pretty simple actually. Covered - hand salute. Uncovered - attention. The observation about the Pledge of Allegiance is a different matter. I can't recall ever having encountered the question in almost 30 years of active duty! So now I'm a retiree and veteran and encouraged to salute at appropriate times while in civilian clothes. Whenever I'm at a ceremony outdoors I usually have a hat and salute the colors. But indoors? I'm used to just standing at attention but then feel I've messed up by not putting my hand over my heart as I would easily do during the pledge. Maybe we shouldn't worry about such stuff so much. Rendering appropriate respect to the event and honors is what is most important. I'm OK with the Marine in this photo.

I would agree with what several people have mentioned here. Since he is wearing his cover he should have saluted.

How about we let the Marines police their own...