I think that much of this thread has shifted away from the OP’s question (Can you be fired in the US without warning?) to a unionization question. The issue of whether you are in a Right to Work state has nothing to do with the OP’s issue. “Right to work” referes to those states where the legislature has enacted a statute which bans closed shops. In other words, you cannot be required to join a union in order to work at a particular job.
To answer the OP, most (if not all) US states follow the employment at will doctrine, with modifications. Employment at will means that both the employer and employee can terminate the employment relationship at any time, for any reason, or no reason. This doctrine is modified to limit the employer’s right to fire the employee in all states that I am aware of. In Illinois (where I practice - IAAL), there are several state statutes and common law doctrines which, when added to the nationally-applicable Federal statutes which come into play. Also, some municipalities have their own ordinances.
First, there are numerous anti-dicrimination laws. You can’t be fired because of your race, gender, or age, for example. This is not a complete list of protected classes. To give one other example, Chicago bars employment discrimination based upon sexual preference.
Second, if you have a contract (including a union collective bargaining agreement), you probably can’t be fired without cause. Most employees do not have contracts, however. For those that do, their rights are governed by what the contract says. Occasionally, courts have held that an employee manual is a contract, but unless your employer is a moron (or has a moron for an attorney), your employee manual will expressly state that it is NOT a contract, for precisely this reason.
Third, you can’t be fired for reasons that conflict with a significant public policy. Sometimes these restrictions have been legislated. In some other circumstances, courts have recognized the public policy. For example, you can’t be fired for reporting your employer for committing a crime (whistleblowing), serving on a jury, or filing a workers’ compensation claim. Interestingly, in Illinois, the one type of an employee who can be fired for whistleblowing is an attorney, because of the higher duty of loyalty that we owe clients. (This applies to situations where the employer is the client, or in other words, where the attorney is part of an in-house staff.)
Lastly, in practice, it is considered to be good form to give at least two weeks notice (in the case of the employee) or severance pay (in the case of the employer) even if there is no legal right to do so.