The former world champion announced his retirement from the ring with immediate effect in January and the promoter believes that The Gypsy King should remain true to his word and leave the sport behind.
Arum told Sky Sports: "I think Tyson Fury owes it to himself and his family to remain happily retired.
"Not from life, because I believe he has a lot in front of him. He is a good communicator and is an entertainer, but I don't think he should step in the ring again because I think he's had enough.
"Fortunately, in the last few years of his career he earned so much money his family or himself is never going to want for money. I would advise him not to ever go back in the ring, but to make himself useful as a spokesperson because he can communicate extraordinarily well.
"People are intrigued by him and I think he can do a lot of good."
Fury's much-discussed fight with Anthony Joshua will never take place should he stay retired but Arum doesn't think the bout is a good enough reason for the British boxer to reverse his decision.
The American promoter said: "As you know, I'm not a Brit, so I'm not as fascinated by an Anthony Joshua vs Tyson Fury fight as a Brit may be.
"So it doesn't move me that he's walking away from boxing and walking away from that fight.
"Tyson is a very outward kind of person. He communicates well and reaches out and expresses what he thinks. But that being said, I really don't know what deep down he's going to do because I don't think he really does."