We present detailed ab initio study of structural and magnetic stability of a Fe monolayer on the fcc(001) surface of iridium. The Fe monolayer has a strong tendency to order antiferromagnetically for the true relaxed geometry. On the contrary an unrelaxed Fe/Ir(001) sample has a ferromagnetic ground state. The antiferromagnetism is thus stabilized by the decreased Fe-Ir layer spacing in striking contrast to the recently experimentally observed antiferromagnetism of the Fe/W(001) system which exists also for an ideal bulk-truncated, unrelaxed geometry. The calculated layer relaxations for Fe/Ir(001) agree reasonably well with recent experimental low-energy electron diffraction data. The present study centers around the evaluation of pair exchange interactions between Fe atoms in the Fe overlayer as a function of the Fe/Ir interlayer distance which allows for a detailed understanding of the antiferromagnetism of a Fe/Ir(001) overlayer. Furthermore, our calculations indicate that the nature of the true ground state could be more complex and display a spin spiral like rather than a c(2x2)-antiferromagnetic order. Finally, the magnetic stability of the Fe monolayer on the Ir(001) surface is compared to the closely related Fe/Rh(001) system.