One repents of sin. Why should he repent of being honestly, totally, dead wrong? (Or freepin’ bats.) He could apologize for being mistaken, but that’s different.
Unless it was a swindle, of course, but he seemed to believe in it himself.
“But concerning that day and hour no one knows, not even the angels of heaven, nor the Son, but the Father only. For as were the days of Noah, so will be the coming of the Son of Man.
For as in those days before the flood they were eating and drinking, marrying and giving in marriage, until the day when Noah entered the ark, and they were unaware until the flood came and swept them all away, so will be the coming of the Son of Man. Then two men will be in the field; one will be taken and one left. Two women will be grinding at the mill; one will be taken and one left.
Therefore, stay awake, for you do not know on what day your Lord is coming. But know this, that if the master of the house had known in what part of the night the thief was coming, he would have stayed awake and would not have let his house be broken into. Therefore you also must be ready, for the Son of Man is coming at an hour you do not expect.” - Matt 24
It is a very weighty thing to speak for God. Camping misrepresented God Almighty to billions of people. He claimed that the Bible guaranteed the world would end on May 21st, even though the Bible said no such thing. Even though Jesus Himself said that only the Father knew when that day was, Camping claimed to know better. Camping is a false prophet, not just mistaken, and he led many people astray and has given cause for the world to mock Christianity.
Not only that, but repentance is between a man and God. In no way should anyone be asked to repent to their fellow man. First, it would do no good, and second, the spectacle of calling for public humiliation is disgusting.
Because to be "honestly wrong" he had to:
(1) Ignore a significant amount of what is written in the Bible (including the "no man knows the day" teaching by Jesus). This is tantamount to ripping pages out of the Bible, because you don't like what they say.
(2) He had to twist and bend scripture in other areas to support his prophecy. This includes being dogmatic about verses--demanding that there is only once correct intrepretation of a verse: the intrepretation that supports YOUR position.
(3) The Bible bluntly states that those that hold themselves out to be religious teachers will be held to a higher standard by God come judgement day.
(4) This is the second time he has made the exact same "honest mistake." As the old saying goes: once is an accident, the second time is a pattern. And the "honest mistake" he keeps making is placing his own, private interpretation of scripture above what the text of the Bible clearly states--states in such a clear manner that even a child can understand it: no man knows the hour or the day.