Daniel Pipes' response:
I do not accept the premise of this question; I would ask "Why does the world resent America?" A country that is truly hated would not be under seige from illegal immigration, its popular culture would not dominate, and its model of government and economy increasingly emulated.
But accepting your question as posed, it suggests that the United States finds itself in a position comparable to the Arab oil sheikhs of the 1970s and Japan during the 1980s. This points to an answer: in each of these three cases, the offending party enjoyed a power that others perceived as overweeining, somewhat illegitimate, and threatening.
The resentment against Americans presumably will continue until their power diminishes – note how little animus is directed toward the Japanese these days. Short of becoming less dominant, Americans can do little to reduce the hostility directed their way.
That said, acting with care and modesty, making concessions where these do not harm vital interests, is a good idea and could go some way to improve the general mood.