  Euclid is poised to survey galaxies across a cosmological volume of
unprecedented size, providing observations of more than a billion objects
distributed over a third of the full sky. Approximately 20 million of these
galaxies will have their spectroscopy available, allowing us to map the 3D
large-scale structure of the Universe in great detail. This paper investigates
prospects for the detection of cosmic voids therein and the unique benefit they
provide for cosmology. In particular, we study the imprints of dynamic and
geometric distortions of average void shapes and their constraining power on
the growth of structure and cosmological distance ratios. To this end, we made
use of the Flagship mock catalog, a state-of-the-art simulation of the data
expected to be observed with Euclid. We arranged the data into four adjacent
redshift bins, each of which contains about 11000 voids and estimated the
stacked void-galaxy cross-correlation function in every bin. Fitting a
linear-theory model to the data, we obtained constraints on $f/b$ and $D_M H$,
where $f$ is the linear growth rate of density fluctuations, $b$ the galaxy
bias, $D_M$ the comoving angular diameter distance, and $H$ the Hubble rate. In
addition, we marginalized over two nuisance parameters included in our model to
account for unknown systematic effects. With this approach, Euclid will be able
to reach a relative precision of about 4% on measurements of $f/b$ and 0.5% on
$D_M H$ in each redshift bin. Better modeling or calibration of the nuisance
parameters may further increase this precision to 1% and 0.4%, respectively.
Our results show that the exploitation of cosmic voids in Euclid will provide
competitive constraints on cosmology even as a stand-alone probe. For example,
the equation-of-state parameter $w$ for dark energy will be measured with a
precision of about 10%, consistent with previous more approximate forecasts.
