First of all, congratulate your surgeon on his or her honesty; this CAN occur in any tummy tuck patient, and is certainly more likely when an umbilical hernia is repaired during tummy tuck.
The opening in the abdominal wall for your umbilicus (and its blood supply) was stretched, allowing abdominal fat, or if large enough, even abdominal contents such as small intestine to herniate (protrude) through this stretched opening. This is the definition of "umbilical hernia." After pregnancy and childbirth, small umbilical hernias present as "outie" belly buttons. Larger hernias can become symptomatic or even dangerous if small intestine becomes trapped and kinked off in a strangulated hernia. Closing this hernia opening to prevent the protrusion is what it takes to return your umbilicus to an "innie," but this can also compromise the circulation to the umbilical skin.
Gray or bluish discoloration indicates circulatory compromise, and can lead to dead umbilical skin. I see all of my tummy tuck patients the next day to check this, and add nitropaste or other considerations to maximize circulation and diminish the risk of dead umbilical skin. But sometimes this occurs despite our best efforts (or if the patient is a smoker or exposed to second-hand smoke--nicotine is a potent vasoconstrictor and can single-handedly kill skin that has marginal circulation).
At 9 days post-op, your umbilcus is likely either dead or alive (not "dying"), since by now the circulation (or lack thereof) has already declared itself. If dead, there is nothing to do but keep the living tissues from becoming infected (antibiotics, careful wound care, and timely follow-up visits) and allow things to heal. Surgical debridement may be necessary if the dead skin becomes infected. You will ultimately develop a scar in this area that in many cases actually resembles a normal umbilicus quite well. In other cases, surgical creation of a "new" umbilicus can be carried out by your surgeon.
If the belly button circulation was compromised at the beginning of your post-op period and is now recovering, keeping the area clean and protected (and perhaps again considering antibiotic coverage to minimize the risk of infection) will allow the tissues to heal as rapidly as possible.
Perhaps your surgeon is "breaking it to you gently" if 9 days is the first time you have been seen post-op. If the belly button is black, dry, and firm, it is not "dying," it is already dead. Read two paragraphs previously. If the skin is dead, it cannot be revived, but all measures should be taken to minimize adjacent tissue loss or infection of skin or muscle repair sutures that could lead to other problems. See your surgeon frequently and follow wound care protocols carefully. This may take a bit longer than you or your surgeon initially planned, but ultimately things work out just fine in the vast majority of cases. Best wishes!