Is RSA provably secure in the sense of Douglas Stinson's ``provable security''?
Every cryptosystem is "provably secure" under at least one hardness assumption: the assumption that it cannot be broken. Hence, the only question which matters is whether a cryptosystem is provably secure under a well-known and well-studied assumption.

This is kind of the case for RSA, but in a somewhat unsatisfying way: the IND-CPA security of RSA with appropriate padding can be reduced to the RSA problem (extracting $e$-th roots mod $n$), which is a well-studied assumption... But only because this is the one underlying the RSA cryptosystem, and we have been using the cryptosystem a lot in the past decades.

Something much more satisfying would be to reduce it to a problem which had long been studied before, in different context; the most natural such candidate is indeed factoring. Unfortunately, we do not know of a reduction from RSA to the hardness of factoring integers - in fact, we even have proofs that it will be hard to find such a reduction (in a well defined sense).

We have, however, many other cryptosystems which can be reduced to factoring; the earliest example of this kind is the cryptosystem of Rabin.