  We construct a model structure on the category $\mathrm{DblCat}$ of double
categories and double functors, whose trivial fibrations are the double
functors that are surjective on objects, full on horizontal and vertical
morphisms, and fully faithful on squares; and whose fibrant objects are the
weakly horizontally invariant double categories.
  We show that the functor $\mathbb H^{\simeq}\colon \mathrm{2Cat}\to
\mathrm{DblCat}$, a more homotopical version of the usual horizontal embedding
$\mathbb H$, is right Quillen and homotopically fully faithful when considering
Lack's model structure on $\mathrm{2Cat}$. In particular, $\mathbb H^{\simeq}$
exhibits a levelwise fibrant replacement of $\mathbb H$. Moreover, Lack's model
structure on $\mathrm{2Cat}$ is right-induced along $\mathbb H^{\simeq}$ from
the model structure for weakly horizontally invariant double categories.
  We also show that this model structure is monoidal with respect to B\"ohm's
Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the
weak equivalences with fibrant source as the double functors which admit a
pseudo inverse up to horizontal pseudo natural equivalence.
