My question comes from two names.
For a finite (topological) simplicial complex $\Delta$, we have its Stanley-Reisner ring $k[x_1,\dots,x_n]/I_\Delta$. This ideal is homogeneous so we can take the $\mathrm{Proj}$ construction getting a projective variety (we assume $k$ is algebraically closed).
Because $\Delta$ is set to be finite, we have a polytope that is simplicial. Then we have a toric variety over $k$ associated to this polytope.
Do these constructions give us the same varieties? (If the answer is yes, we can know that this variety is simplicial as well.) (For this variety being simplicial, I might be wrong.)
The components of $\mathbf{P}(\Delta) = \operatorname{Proj} k[x_1,...,x_n]/I_{\Delta}$ are in 1-1 correspondence with the maximal faces of $\Delta$, so you never get a toric variety this way, except in the case of $\Delta$ being a simplex (in which case you get $\mathbf{P}(\Delta) = \mathbf{P}^n$).
Now, I'm assuming you're talking about $\Delta$ being the boundary of a simplicial polytope. In that case, there's not much of a connection between the Stanley-Reisner (projective) scheme, and the toric variety $X_P$. However, if all the facet simplices of $P$ are unimodular, i.e. of volume $1$ (or dilations of unimodular simplices), then we do in fact have that $\mathbf{P}(\Delta)$ is isomorphic to the torus-invariant anti-canonical divisor $-K_{X_P} = \sum_{F \text{ facet}} D_F \subseteq X_P$.