The motorist broke down near junction 30 on the motorway heading northbound, causing long queues to form back.
The man who was ticketed for leaving the vehicle in a dangerous condition then went on to have an argument about charges for recovering his vehicle.
Police said there were multiple services where fuel was available before the driver's car ran out.
Derbyshire Roads Policing Unit tweeted at 8.48pm on Saturday, September 5: "M1. This is the tailback and disruption caused by a motorist running out of fuel in a live lane. Proceeds to have an argument with HighwaysEMIDS about recovery fees. "
They added: "Options for fuel prior to this happening near J30 Northbound. J25 to J26 Trowell Services J28 to J29 Tibshelf Services J29A Duckmanton Services Plus filling stations within a mile of J26 and J28. Thanks."