If a tenant moves out without paying rent can you serve him with an eviction notice for unpaid rent even though he already moved out? The purpose is to put an eviction on his credit if he does not pay.

If you have to ask these questions you didn't prepare yourself for being a landlord.

How do you know they moved out? Also, the answer to your question depends on where the rental property is. Not sure how anyone can ask a question about law without giving, at least a general area so we can provide the applicable law.

If there is rent due, you serve them with a pay or quit notice. The amount of time you give them depends on the local laws. If they don't pay in full by the due date then you go to court for an eviction order.

Agree , the word evict means to "to put (a tenant) out by legal process" if you go with Webster's definition. You can't evict someone who isn't there. How have you determine they are gone and the property is abandoned?

What state are you in?

Do you have the security deposit? Usually, you can take it out of that with a note to the past tenant explaining the reason for withholding.