Hello and welcome,
Signing a written warning generally indicates that the employee reviewed it, not that they agree with it, unless there is language in the document to that effect. So it does not matter whether the employee signs it or not, except the signature indicates they have seen the document.
Unless there is a contract stating otherwise, employers do typically have the discretion to provide performance reviews for employees, even if they are incorrect, and the employee would not have recourse.
There are exceptions, such as when the negative review is motivated by a discriminatory intent based on race, gender, age, or other protected status.
If an exception does not apply, there would normally be no legal recourse by the employee.
I would like to note though that when an employee retains an attorney to communicate with the employer, sometimes the fact that an attorney is involved motivates the employer to act more reasonably, so you may want to consider that option even if there are no legal grounds to challenge the employer's action.
I hope you found my answer helpful, even if the law is not in your favor. I would be happy to clarify my answer if anything is not clear or you are looking for additional information. All the best to you!