Can I publicly call my company "example inc" or does it have to be named without the "inc." label?
No, you cannot add the word 'Inc' to a trade name.
I take it that you want the company to use a name that does not belong to the company, by adding 'Inc' to the end of a business name that does belong to the company. For instance, your company is called 'X Computer Inc', the company holds a registered business name 'X', and you want it to use the name 'X Inc'.
An S corp is not something that can do business. 'S corp' and 'C corp' are federal tax designations. You have some kind of corporation under state law that has a specific name ending in 'Inc'.
'Inc' stands for 'incorporated'. A registered business name ('fictitious business name', 'doing business as' or 'DBA') does not represent incorporation. There is no reason why you could append 'Inc' to a business name.
What are the consequences of your company calling itself X Inc if that is not its name?
Most states' laws contain a requirement to register a trade name if you are doing business under a made-up name, e.g. California's Business and Professional Code, ss 17900ff. Permissible business names include an individual's name and a corporation's name. Sanctions depend on the jurisdiction and include fines, injunctions (breach of which would be contempt and put you in prison) or a refusal by the state to enforce any contract you enter into under the unregistered name.
You cannot add 'Inc' to the end of your registered business name any more than you can add 'of America' or 'Purple Monkey Dishwasher' to it. By doing so, you would expose yourself to penalties under state law for trading under a false name. In California, for example, see in particular the Business and Professional Code s 17900(b)(3).