$\newcommand{\Z}{\mathbb{Z}}\newcommand{\G}{\mathbb{G}}$

I emailed Johan de Jong, and he sent me the following, which I reproduce with his permission (lightly edited to only keep mathematical content).
Gabber's proof is not written up anywhere, from what de Jong told me.

The argument is different because I lectured about the proof from my
write-up in front of Gabber and he then subsequently told me it is
different. After the lecture he also explain his proof. Roughly what
he does is two things:

(1) Prove it for regular quasi-projective schemes. This involves
carefully choosing a maximal order over the scheme which is Azumaya by
modifying along an ample divisor (on some affine open you can already
do the thing).

(2) Prove the following amazing theorem: Suppose that $X$ is a scheme
which has an ample invertible sheaf. Suppose that $\alpha$ is a
cohomology class in $H^i(X, \Z/n\Z)$ for some $n > 0$ and since $i \ge 0$.
Then one can find a quasi-projective scheme $Y$ smooth over $\Z$ and a
morphism $X \to Y$ and a $\beta \in H^i(Y, \Z/n\Z)$ which pulls back to
$\alpha$. Moreover, the same thing can be done with torsion classes in
$H^2(X, \G_m)$. To do this you may immediately assume that $X$ is of finite
type over $\Z$ by a limit argument. Then you can embed $X$ into some smooth
$P$, e.g., projective space. Now on an open you can sort of extend the
class (after replacing $P$ by a blow up in closed subscheme and maybe an
etale neighbourhood). Then, and this was all explained to me at a
party later in the day [...] you kind of keep blowing up until $\alpha$ extends.