Context. A gravitational wave (GW) passing through an astrometric observer causes periodic shifts of the apparent star positions measured by the observer. For a GW of sufficient amplitude and duration, at a suitable frequency, these shifts might be detected with a Gaia-like astrometric telescope. Aims. This paper is aimed at making a detailed analysis of the effects of GWs on an astrometric solution based on Gaia-like observations, which are one-dimensional and strictly differential between two widely separated fields of view, following a prescribed scanning law. Methods. We present a simple geometric model for the astrometric effects of a plane GW in terms of the time-dependent positional shifts. Using this model, we discuss the general interaction between the GW and a Gaia-like observation. Numerous Gaia-like astrometric solutions have been computed, taking as input simulated observations that include the effects of a continuous plain GW with constant parameters and periods ranging from ∼50 days to 100 years. The resulting solutions have been analysed in terms of the systematic errors on astrometric and attitude parameters, as well as the observational residuals. Results. We found that a significant part of the GW signal is absorbed by the astrometric parameters, leading to astrometric errors of a magnitude (in radians) comparable to the strain parameters. These astrometric errors are generally impossible to detect because the true (unperturbed) astrometric parameters are not known with a corresponding level of accuracy. The astrometric errors are especially large for specific GW frequencies that are linear combinations of two characteristic frequencies of the scanning law. Nevertheless, for all GW periods smaller than the time span covered by the observations, significant parts of the GW signal also go into the astrometric residuals. This fosters the hope for a GW detection algorithm based on the residuals of standard Gaia-like astrometric solutions.