No, there is no law that states that an employee must sign a write-up form, however, employers may need proof that an employee has received it. In many cases, employees refuse to sign written warnings because they do not agree with them.
To avoid this situation, include a line on the write-up stating that the employee acknowledges it, but does not necessarily agree with its contents, and provide an area for the employee to sign. If the employee still refuses to sign, you can ask them to write a rebuttal and attach it to the write-up form.