No.
I do agree he has an large ego ( as do many politicians, businessmen etc at his level). I doubt if he has fragile self esteem. Fighting back doesn't necessarily mean that.
I don't know if he cares how much he is admired. At least any more than most people do.
From the Mayo Clinic
"Narcissistic personality disorder is a mental disorder in which people have an inflated sense of their own importance, a deep need for admiration and a lack of empathy for others. But behind this mask of ultraconfidence lies a fragile self-esteem that's vulnerable to the slightest criticism."
"A narcissistic personality disorder causes problems in many areas of life, such as relationships, work, school or financial affairs. "
He has done very well in business and other endeavors. He has been divorced a couple of times but seems to have a good relationships with his family (including in-laws and ex wives).