Power Politics: Tragedies and Transitions
THE question at hand is simple and profound: will China rise peacefully? My answer is no. If China continues its impressive economic growth over the next few decades, the US and China are likely to engage in an intense security competition with considerable potential for war.
Mearsheimer is simply applying his Offensive Realist theory ("The ultimate goal of every great power is to maximise its share of world power and eventually dominate the system.") to a rising China, as Logan points out in greater detail. Mearsheimer's conclusion (China is rising and America doesn't want a peer competitor in Asia) isn't particularly novel. This has all been said before and seems to me to be entirely to simplistic to be interesting, much less enlightening.
Power Transition Theory
A much more interesting analysis, in my opinion, is that of Jacek Kugler and Ronald Tamen. Regional Challenges: China's Rise to Power (PDF) looks at China's rise through the eyes of Power Transition theory, which is explained in detail here.
Simply put, Realists tend to disagree as to whether the international system is most stable with one, two, or multiple powers. Power Transition theorists would argue the number is irrelevant. Wars result from shifts in the distribution of power. When the hierarchy of power in a region is challenged and the rising power is dissatisfied with the international system, war is likely to result.
Since most experts expect China to surpass the US economically near the middle of this century, the answer to Dr. Mearsheimer's question above can be found in to what degree China is satisfied with the current international system.