You certainly do. Skytrain belongs to the public transport authority, ICE is Deutsche Bahn. To avoid the extra ticket you can take the S11 (which belongs to Deutsche Bahn) from the terminal station below the airport to Duesseldorf main station and board your ICE there. This is to my belief by far more convenient at least for any ICE journey towards the south than changing twice (at DUS Airport and Düsseldorf Main).

Only exceptions ticketwise:

You hold a Bahncard and got a free "city-ticket",

https://www.bahn.com/i/view/GBR/en/p...y-ticket.shtml

you bought an add-on with your ICE ticket called "city mobil"

https://www.bahn.de/p_en/view/offers...itymobil.shtml

or you hold a ticket good for the public transport authority's services only (which includes several Deutsche Bahn services but not the ICEs and ICs).