Eales is stepping down after a cancer diagnosis.
Al-Rumayyan said: "All of us who are connected with Newcastle United are very saddened by this news. Our thoughts are with Darren and his family and we send him every good wish for a full recovery.
"On a personal level, Darren has been a great partner to me and to our fans, players, ownership and staff. On behalf of the club and its owners, I want to thank Darren for his service and dedication to Newcastle United. He has made a significant contribution to the club's progress during his time as CEO, helping to build a strong foundation for Newcastle United's success. We fully understand the difficult decision he has made.
"We will work closely with Darren to hand over the CEO role at the appropriate moment following a carefully considered search process, and he will continue to have the full support of the board and the Newcastle United family now and in the future."