Do physicists ever expect to be able to derive the fundamental constants of nature from theory? For example, if string theory or some other theory unites the four forces, would the theory be considered complete if it relies on these measured constants, or would a true theory of everything (TOE) require that these constants come out of the theory itself?
A physical theory needs many experimental inputs - form of equations and constants in them. A theory claiming to be able to calculate all constants in its equations is mathematics, not physics. A recent take on it is presented by S. Weinberg here.
In this case, opinions diverge.
Some people claim that only two constants are needed: the speed of light $c$ and the string length $\lambda_s$.
Other think that you also need $h$.
You might want to have a look at: http://arxiv.org/abs/physics/0110060