Legal Question in Employment Law in Washington

Unless you have a contract (either a collective bargaining agreement or an individual contract), you can be terminated for any reason or for no reason at all. Your employer can terminate you based on hearsay. You won't have legal recourse against the employer, unless you can show that the employer used the hearsay as a pretext, and the real reason was an illegal one (discrimination for an illegal reason or retaliation for exercising your statutory rights, for example).

However, you will likely qualify for unemployment compensation unless the employer can show persuasive evidence of misconduct on your part.

Also, if the hearsay was false, and you know who was spreading the false rumor, you may have a defamation claim against that person. It is hard to say based on the minimal facts in your post, so I would encourage you to talk with an employment law attorney with whom you can discuss the matter in more detail.