References
Sketch of a proof
Etale descent
and is also generalized in the paper of Rosenschon and Ostvær.
[Edit]
Sketch of a proof, part 2
Let me give more details on the proof, now that I understand the details a little better. The main ingredients are
Nisnevich descent for perfect complexes, as a prestack of stable infinity-categories $X \mapsto Perf(X)$.
Compact generation of $D(X)$ by the perfect complexes, and also for the version "with support" on a closed subscheme $Z \subset X$, i.e. for the full subcategory $D_Z(X)$ of complexes that vanish on $X-Z$.
The Thomason-Neeman localization theorem: an exact sequence of stable infinity-categories induces a fibre sequence of K-theory spectra.
Then the proof is as follows. Let $j : U \hookrightarrow X$ be an open immersion and $p : Y \to X$ an etale morphism defining an elementary Nisnevich square
$$\require{AMScd}
\begin{CD}
W @>>> Y \\
@VVV @VV{p}V \\
U @>>{j}> X
\end{CD}$$
We want to show that the square
$$\require{AMScd}
\begin{CD}
K(X) @>{j^*}>> K(U) \\
@V{p^*}VV @VVV \\
K(Y) @>>> K(W)
\end{CD}$$
is a homotopy cartesian square of connective spectra. (In the noetherian case, Morel-Voevodsky showed that Nisnevich descent is equivalent to this Brown-Gersten-style excision property; in the non-noetherian case, this should really be taken as the "correct" definition of the Nisnevich topology.)
For this it is sufficient to show that there is an equivalence on homotopy fibres.
2) and 3) imply that the homotopy fibres are given by $K_{X-U}(X)$ and $K_{Y-W}(Y)$, respectively.
Then the equivalence follows from the fact that there is already an equivalence
$$ Perf_{X-U}(X) \stackrel{\sim}{\longrightarrow} Perf_{Y-W}(Y) $$
at the level of perfect complexes, by 1).
This proof works for quasi-compact quasi-separated schemes. By the way, the same proof works for (qcqs) derived schemes. By far the most non-trivial part of the proof is 2), which was established in [B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599]. A proof of 1), attributed to Drinfeld, is given in [D. Gaitsgory, Notes on geometric Langlands: Quasi-coherent sheaves on stacks, pdf].
