For medical you need the actual document
Nope. You can get a faxed certificate that you have a medical (not a copy of the medical) from the FAA in Oklahoma City. This document acts as 'proof of medical'.
Happened to me once - arrived in the US and realized I had left the medical in the UK. A couple of phone calls later I had the above doc and was good to go.
Btw, the friendly lady at the FAA advised strongly against simply doing another medical (I only needed a third class and thought this might be the quicker way of dealing with the situation), as - in her words - this would 'raise red flags' with the FAA medical division; mine was at the time still valid for about a year.