There is no way it is true in general, but there are results in this direction nevertheless: Theorem 3.1 in this paper of Voevodsky establishes (a kind of fully) faithfulness for normal schemes of finite type over a number field, in the spirit of a conjecture of Grothendieck (in his letter to Falting, an English translation of which is published in Geometric Galois actions, 1, ser. London Math. Soc. Lecture Note Ser. Vol. 242, 1997, and in his Esquisse d'un programme).

In particular, if $X$ and $Y$ are normal schemes of finite type over a number field $K$ (in fact any field of finite type of characteristic zero) and if their small étale topoi are equivalent over the small étale topos of $K$, then $X\cong Y$ as $K$-schemes.

There are also Torelli-like theorems saying that one can sometimes determine the isomorphism class of a scheme from its topological space only (i.e. from the 0-localic topos associated to its étale topos). This paper of János Kollár, Max Lieblich, Martin Olsson, Will Sawin shows that, for $K$ an uncountable algebraically closed field of characteristic zero $K$, one can reconstruct a normal geometrically integral proper scheme of dimension at least 2 from its underlying topological space. Their results also prove, for $K$ an uncountable algebraically closedfield of characteristic zero, one can reconstruct a proper normal $K$-scheme of dimension at least 2 from its category of constructible abelian étale sheaves.

(pseudo)functortaking schemes to their étale toposes fully faithful? Then Jens's answer shows the answer is no. $\endgroup$