The answer is yes. It is theorem [2.13] of the following paper of Prasad and Raghunathan:

Prasad, Gopal; Raghunathan, M. S. Cartan subgroups and lattices in semi-simple groups. Ann. of Math. (2) 96 (1972), 296–317.

There is also a lot of information in this paper:
http://www.math.bgu.ac.il/~barakw/papers/clorbit.pdf

Note that diagonalizable copies of $\mathbb{Z}^{n-1}$ in $\Gamma$ correspond to closed orbits for the action of the full diagonal subgroup of $SL(n,\mathbb{R})$ on $SL(n,\mathbb{R})/\Gamma$.

This is related to the Margulis conjecture which (with some caveats) states that the closure of any orbit of the full diagonal on $SL(n,\mathbb{R})/\Gamma$ is algebraic, i.e. is itself the closed orbit of some subgroup. This conjecture is the biggest open problem in homogeneous dynamics (and in particular implies the Littlewood conjecture in number theory).