I don't think "whine like a child" is a fair assessment. He wasn't upset about eating his vegetables or going to the dentist. He was faced with his impending death, in a sense -- the Doctor would live on, but the personality and nature he had now would be changed into something different, and in a way the person he was would die. And this wasn't a death that came on him suddenly, but one he'd been warned about well in advance and had time to dwell on. I hardly think it's uncommon for, say, people with terminal diseases to go through phases of anger and bitterness, to rail against the universe at the unfairness of it all, and those of us who've never been there have no business judging them for it.