Galaxy proto-clusters are receiving increased interest since most of the processes shaping the structure of clusters of galaxies and their galaxy population happen at the early stages of their formation. The Euclid Survey will provide a unique opportunity to discover a large number of proto-clusters over a large fraction of the sky (14 500 deg2). In this paper, we explore the expected observational properties of proto-clusters in the Euclid Wide Survey by means of theoretical models and simulations. We provide an overview of the predicted proto-cluster extent, galaxy density profiles, mass-richness relations, abundance, and sky-filling as a function of redshift. Useful analytical approximations for the functions of these properties are provided. The focus is on the redshift range z = 1.5-4. In particular we discuss the density contrast with which proto-clusters can be observed against the background in the galaxy distribution if photometric galaxy redshifts are used as supplied by the ESA Euclid mission together with the ground-based photometric surveys. We show that the obtainable detection significance is sufficient to find large numbers of interesting proto-cluster candidates. For quantitative studies, additional spectroscopic follow-up is required to confirm the proto-clusters and establish their richness.