I actually disagree with @Qmechanic: my answer is no, half integer spin cannot be derived from the path integral.
That massive particles come in discrete spins of multiples of half integer values of $\hbar$ is due to two postulates: a system is represented by a state in a Hilbert space (in particular, single particle states live in a Hilbert space) and, separately, relativity, that the universe is symmetric under the 4D Poincaré group IO(1,3). Neither of these postulates is related in any way to correlation functions and/or the time evolution of the system.
To expand slightly, Wigner proved that symmetry transformations are effected by linear, unitary operators on the Hilbert space. (Or anti-linear, anti-unitary op's.) Thus single particle states live in irreducible representations of the symmetry group. When one works out the irreps of the Poincaré group, one finds that massive particles have spins that are multiples of half integers times $\hbar$. (See, e.g., Ch. 2.5 of Weinberg's QFT book.)
That correlation functions are given by the path integral <=> time evolution is determined by the Schroedinger equation is a totally separate postulate. This postulate holds independent of the spacetime symmetries of the universe; in particular, this postulate is agnostic as to whether the universe is governed by non-relativistic physics (the Galilean group) or relativistic physics (the Poincaré group).
I do agree with @Qmechanic that there is a relationship between the fundamental unit of action, which normalizes and makes dimensionless the argument of the exponential in the path integral, and the fundamental unit of angular momentum: these two quantities have the same units, kg m$^2$/s, so we shouldn't be surprised that they're one and the same.
Edit: As was pointed out by @Ryder Rude, one can derive half integer spin non-relativistically from the rotation group SO(3); i.e. the Galilean group will still have half integer spin. But the point still stands that the half integer spin comes from the combination of quantum mechanics requiring a Hilbert space for states to live in and the states themselves living in an irrep of the symmetry group; i.e. there's nothing to do with time evolution and/or path integrals. It's worth remembering further that for 2 spatial dimensions one has anyons, so the half integer nature of spin is a quirk of the number of spatial dimensions in the problem.