I remember hearing someone answer "if the brain was simple enough that we could understand it, it would be so simple that we wouldn't" to that question.
IC design is largely a process of simulating and specifying the next generation of integrated circuits.
So in the strict sense, not only can a computer compute itself, but it can compute its (more capable) successor generation.
Practically, this now occurs in large server farms, so it takes a large number of computers to compute themselves and/or progeny. But this is very much what happens.
Can a computer compute itself?