Yes, this is a private site and Arn can do with it whatever he chooses. If her wakes up tomorrow morning and decides to ban everyone who's username starts with the letter G then he is free to do so.
However, it is not in his best interests to do so, since this will obviously not be taken well by the users. Less users, means less ad clicks, means less revenue.
Thus, he has created Rules, so users know what is expected of them, and has appointed moderators to enforce such rules.
Now, in this case, the user has not broken any of the Rules as written, but apparently done something that is unacceptable to Arn and/or the moderators.
The first interesting question is this: Do the moderators have from Arn a set of additional rules (such as this one) to enforce that are not in the official rules, or did the moderator simply remember Arn saying that and thus chose to warn the user about it?
Again, Arn can choose to do whatever he likes, but it is in his best interests to have a userbase that knows where they stand.