  We construct a model structure on the category $\mathrm{DblCat}$ of double
categories and double functors. Unlike previous model structures for double
categories, it recovers the homotopy theory of 2-categories through the
horizontal embedding $\mathbb{H}\colon2\mathrm{Cat}\to\mathrm{DblCat}$, which
is both left and right Quillen, and homotopically fully faithful. Furthermore,
we show that Lack's model structure on $2\mathrm{Cat}$ is both left- and
right-induced along $\mathbb{H}$ from our model structure on $\mathrm{DblCat}$.
In addition, we obtain a $2\mathrm{Cat}$-enrichment of our model structure on
$\mathrm{DblCat}$, by using a variant of the Gray tensor product.
  Under certain conditions, we prove a Whitehead theorem, characterizing our
weak equivalences as the double functors which admit an inverse pseudo double
functor up to horizontal pseudo natural equivalence. This retrieves the
Whitehead theorem for 2-categories.
  Analogous statements hold for the category $\mathrm{wkDblCat}_s$ of weak
double categories and strict double functors, whose homotopy theory recovers
that of bicategories. Moreover, we show that the full embedding
$\mathrm{DblCat}\to\mathrm{wkDblCat}_s$ is a Quillen equivalence.
