Do I need to have a physical copy of a green card whilst driving abroad?

Yes. Under current international rules, Green Cards are physical paper documents printed on green paper. You may be required to show the physical document at the border when entering the EU, but this will be a decision for the border authorities to take. You may also be subject to police checks while driving abroad and you will also need to be able to present the document at the scene if you are involved in an accident.