  The delay time distribution of neutron star mergers provides critical
insights into binary evolution processes and the merger rate evolution of
compact object binaries. However, current observational constraints on this
delay time distribution rely on the small sample of Galactic double neutron
stars (with uncertain selection effects), a single multimessenger gravitational
wave event, and indirect evidence of neutron star mergers based on $r$-process
enrichment. We use a sample of 68 host galaxies of short gamma-ray bursts to
place novel constraints on the delay time distribution and leverage this result
to infer the merger rate evolution of compact object binaries containing
neutron stars. We recover a power-law slope of $\alpha = -1.83^{+0.35}_{-0.39}$
(median and 90% credible interval) with $\alpha < -1.31$ at 99% credibility, a
minimum delay time of $t_\mathrm{min} = 184^{+67}_{-79}~\mathrm{Myr}$ with
$t_\mathrm{min} > 72~\mathrm{Myr}$ at 99% credibility, and a maximum delay time
constrained to $t_\mathrm{max} > 7.95~\mathrm{Gyr}$ at 99% credibility. We find
these constraints to be broadly consistent with theoretical expectations,
although our recovered power-law slope is substantially steeper than the
conventional value of $\alpha = -1$, and our minimum delay time is larger than
the typically assumed value of $10~\mathrm{Myr}$. Pairing this cosmological
probe of the fate of compact object binary systems with the Galactic population
of double neutron stars will be crucial for understanding the unique selection
effects governing both of these populations. In addition to probing a
significantly larger redshift regime of neutron star mergers than possible with
current gravitational wave detectors, complementing our results with future
multimessenger gravitational wave events will also help determine if short
gamma-ray bursts ubiquitously result from compact object binary mergers.
