Any smooth projective toric variety is rational, in particular simply connected.

Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a complete intersection into a smooth toric variety then $\pi_1(X)=\{1\}$.

In particular, for instance, abelian varieties of dimension at least $3$ cannot be realized as global complete intersections into a smooth toric ambient.

Any smooth projective toric variety is rational, in particular simply connected.

Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete intersection into a smooth toric variety then $\pi_1(X)=\{1\}$.

In particular, for instance, abelian varieties of dimension at least $3$ cannot be realized as global complete intersections into a smooth toric ambient.