Is it illegal to bully an employee?
It is not, unless the bullying is because of race, sex, religion, disability, or other statutory protected characteristic.

Let's use sex as an example. If an employer bullies an employee because of his or her sex, then that can violate Title VII (and similar state laws).

The reason is that this is a form of sex discrimination. It is not because the bullying – by itself – is illegal.

For example, let's look at Montgomery v. ExchangeBase (Ohio Ct App 07/03/2024) https://www.supremecourt.ohio.gov/rod/docs/pdf/8/2024/2024-Ohio-2585.pdf

Miranda Montgomery alleged a long list of what she referred to as bullying by the employer's president and others. No doubt this was unpleasant for her. The problem for her was that she was unable to connect the important dots. She couldn't show that this offensive behavior was BECAUSE of her sex. The offensive behavior by itself was not enough.

In spite of what I've said above, it's possible that OSHA could conclude that some forms of bullying create a hazardous work environment.

Your results may vary. If you have this issue, consult a good employment lawyer.
(Not me, I'm an arbitrator.)