A relativistic method based on the Dirac equation for calculating fully differential cross sections for ionization in ion-atom collisions is developed. The method is applied to ionization of the atomic hydrogen by antiproton impact, as a nonrelativistic benchmark. The fully differential, as well as various doubly and singly differential, cross sections for ionization are presented. The role of the interaction between the projectile and the target nucleus is discussed. Several discrepancies in available theoretical predictions are resolved. The relativistic effects are studied for ionization of hydrogenlike xenon ion under the impact of carbon nuclei.