The magnetic Eden model (MEM) [N. Vandewalle and M. Ausloos, Phys. Rev. E 50, R635 (1994)] with ferromagnetic interactions between nearest-neighbor spins is studied in (d+1)-dimensional rectangular geometries for d=1,2. In the MEM, magnetic clusters are grown by adding spins at the boundaries of the clusters. The orientation of the added spins depends on both the energetic interaction with already deposited spins and the temperature, through a Boltzmann factor. A numerical Monte Carlo investigation of the MEM has been performed and the results of the simulations have been analyzed using finite-size scaling arguments. As in the case of the Ising model, the MEM in d=1 is noncritical (only exhibits an ordered phase at T=0). In d=2 the MEM exhibits an order-disorder transition of second order at a finite temperature. Such transition has been characterized in detail and the relevant critical exponents have been determined. These exponents are in agreement (within error bars) with those of the Ising model in two dimensions. Further similarities between both models have been found by evaluating the probability distribution of the order parameter, the magnetization, and the susceptibility. Results obtained by means of extensive computer simulations allow us to put forward a conjecture that establishes a nontrivial correspondence between the MEM for the irreversible growth of spins and the equilibrium Ising model. This conjecture is certainly a theoretical challenge and its confirmation will contribute to the development of a framework for the study of irreversible growth processes.