We investigate numerically the low-energy properties of an artificial square spin system in which the nanomagnets are physically connected at the lattice vertex sites. Micromagnetic simulations performed on a single square vertex reveal that type-II vertices always have the lowest energy, in sharp contrast with what is found in lattices made of disconnected nanomagnets, for which type-I vertices are the ground-state configuration. The micromagnetic simulations also show that the energy stored at the vertex sites strongly depends on the type of magnetic domain wall formed by the four connected nanomagnets. Interestingly, the energy gap between type-I and type-II vertices can be drastically reduced by varying the geometrical parameters of the nanomagnets, such as their width and thickness. For typical widths and thicknesses achievable experimentally, we find that this energy gap is small enough to consider type-I and type-II vertices as quasidegenerate. Based on the vertex energies provided by the micromagnetic simulations, we compute the thermodynamic properties of the corresponding spin model using Monte Carlo simulations. In some cases, these properties are hardly distinguishable from those of the celebrated square ice model. Our findings then suggest that an ice physics, characterized by a massively degenerate ground-state manifold at low temperature, may be observed experimentally in a simple square lattice of connected magnetic elements. This work thus provides a route to fabricate artificial algebraic spin liquids using a purely two-dimensional geometry.