This really isn't my area, but let me give it a shot.
On the one hand, every element in GL(2,Z)=Aut(Z^2) gives us an automorphism of R^2 that stabilizes Z^2 (here I'm using the standard basis for everything), hence descends to an automorphism of the torus R^2/Z^2. On the other hand, every automorphism of R^2/Z^2 induces an automorphism of pi_1(R^2/Z^2) = Z^2 (this equality is really a specific isomorphism). It seems to me everything here is compatible, and that it shouldn't be too hard to conclude that the isotopy classes of automorphisms (=self-diffeomorphisms?) of R^2/Z^2 lie in one-to-one correspondence with elements of GL(2,Z).
The final observation to make is that an automorphism of R^2/Z^2 preserves the orientation defined by the basis {(1,0), (0,1)} for the lattice iff the corresponding automorphism in GL(2,Z) preserves the orientation in R^2 defined by the basis {(1,0),(0,1)} - i.e., iff the corresponding automorphism in GL(2,Z) has positive determinant <=> has determinant 1 (since everything in GL(2,Z) has determinant +/- 1).
So if by "modular group" you mean group of isotopy classes of orientation-preserving automorphisms, then I believe the above comments show why this group is SL(2,Z).