Not necessarily. Each state has its own statute on what must be included (or may not be included) in a corporation's legal name. As far as I know, every state will permit the use of "Inc.," "Incorporated," "Corporation" or "Corp." in a corporation's name, but many don't require the use of those "indicator words."
Just as an example, Massachusetts requires that a business corporation include in its name "the word 'corporation," 'incorporated,' 'company,' or 'limited' or the abbreviation 'corp.,' 'inc.,' or 'ltd.,' or words or abbreviations of like import in another language..."
And other forms of corporations (religious organizations, schools and colleges, banks, and many others) don't have to include any of the words or abbreviations in that paragraph.
You'll need to be familiar with the naming requirements of the state or states in which the corporations are domiciled.