No. The employee signs it to prove conclusively that they were given the write up. If they refuse to sign it, that is their prerogative. Of course, it is hard for them to improve performance if they do not have a copy of the write up. You can e-mail it to them with a read receipt as another way of proving delivery. Another person would be an alternate way of proof. Some employees are reluctant to sign because they think it is an admission of fault. The way to deal with that is to have them sign a statement that says "I do not necessarily agree with the contents of this writeup. My signature on it is not an admission of fault or liability, but only an acknowledgment that the employer gave me this write up on this date."
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Your premise that the write has legal meaning is wrong. If you're an "at will" employee," meaning no written contract that says you can only be fired for "cause" or "good cause," and if you're not qa union member, your employer doesn't need a reason to fire you, or to give you any notice, or to do a write up before firing you.
If they have policies for progressive discipline leading to firing, they have to follow their own policies, but they don't need to have firing policies in the 1st place.
My colleague is correct that signing doen't prove the truth of the content of the write up, and as a practical matter, you could be considered even more of a troublemaker for refusing to sign.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.