We have investigated the cosmological constraints that can be expected from measurement of the cross-correlation of galaxies with cosmic voids identified in the Euclid spectroscopic survey, which will include spectroscopic information for tens of millions of galaxies over 15 000 deg^2 of the sky in the redshift range 0.9 ≤ z ≤ 1.8. We have done this using simulated measurements obtained from the Flagship mock catalogue, the o fficial Euclid mock that closely matches the expected properties of the spectroscopic dataset. To mitigate anisotropic selection-bias e ffects, we have used a velocity field reconstruction method to remove large-scale redshift-space distortions from the galaxy field before void-finding. This allowed us to accurately model contributions to the observed anisotropy of the cross-correlation function arising from galaxy velocities around voids as well as from the Alcock-Paczynski e ffect, and we studied the dependence of constraints on the e fficiency of reconstruction. We find that Euclid voids will be able to constrain the ratio of the transverse comoving distance D(M) and Hubble distance D(H) to a relative precision of about 0.3%, and the growth rate fσ(8) to a precision of between 5% and 8% in each of the four redshift bins covering the full redshift range. In the standard cosmological model, this translates to a statistical uncertainty ΔΩ(m) = ±0.0028 on the matter density parameter from voids, which is better than what can be achieved from either Euclid galaxy clustering and weak lensing individually. We also find that voids alone can measure the dark energy equation of state to a 6% precision.