OK, I found an article in Wikipedia called Unfair dismissal in the United Kingdom. Now I have something to read and learn about.

But I honestly have never heard of such a law in the US.

I did find an article called Wrongful dismissal. The article contains a disclaimer/warning: "The examples and perspective in this article may not represent a worldwide view of the subject."

It says:

Being terminated for any of the items listed below may constitute wrongful termination:
•        Discrimination: The employer cannot terminate employment because the employee is a certain race, nationality, religion, sex, age, or in some jurisdictions, sexual orientation.
•        Retaliation: An employer cannot fire an employee because the employee filed a claim of discrimination or is participating in an investigation for discrimination. In the United States, this "retaliation" is forbidden under civil rights law.
•        Employee's Refusal to Commit an Illegal Act: An employer is not permitted to fire an employee because the employee refuses to commit an act that is illegal.
•        Employer Not Following Own Termination Procedures: Often, the employee handbook or company policy outlines a procedure that must be followed before an employee is terminated. If the employer fires an employee without following this procedure, the employee may have a claim for wrongful termination.

As you can see, that's a pretty narrow scope. I think I could add a couple more categories of people who are protected, but they too are pretty narrowly defined and don't apply to most situations.

There are separate and detailed sections about the UK and Canada, but it gives very little information about the US per se.

All the discussion in this thread about unions is pretty much irrelevant because it only applies in situations where there is a union and the union has a contract with the employer. Most American workers do not belong to unions. And the original question was about US law.