Being relieved of command in the military can mean a lot of things. People might be relieved of command because they are unfit to lead, or because someone more qualified has been promoted, or because their tour of duty is up, or just because they aren't needed anymore. Even in cases where the officer is being punished (effectively demoted), the circumstances vary quite a lot over the severity of the punishment.
Relieving an officer of a command duty but leaving them on active duty would send the message that the army still thinks the officer has the potential to redeem themselves, but for whatever reason, is not currently fit to lead a command.
However, by also relieving them of their ammunition, his superiors are sending a much stronger message. The message here is actually a combination of all of the things you mentioned, and others. In essence, the army is saying "we no longer trust you to be armed while on base." There's no other way to interpret such an action other than as a punishment for failure to perform their duties properly.
From an image standpoint, this shows that the army has lost faith in the officer, and they are most likely going to be withdrawn from active duty and sent home, likely to be discharged, in short order. (This is effectively what the remainder of Ferrando's speech says: Dowdy didn't "get it" so he's been "fired".) From a practical standpoint, such an action is adding disgrace onto someone who's already humiliated, and likely just lost a lot of current and future salary and pension, and may not be in the best frame of mind to be running around armed.
However, it's also important to note that they confiscated his ammo, not his sidearm. That would be an even bigger disgrace, essentially stripping Dowdy of the thing that defines a soldier. By letting him keep his firearm (unloaded) they're making a symbolic gesture that lets others know that they don't think Dowdy is a bad person, and don't think he did anything "criminally bad". His superiors just don't think he is fit to be in the command position he's in, so they're pulling him out.