Aquatic vegetation in fluvial systems is often characterized by spatial patterning of the plant patches. To investigate the conditions for the formation of vegetation patches, we explore the stability of a uniform flow over a non-erodible bed with a uniform vegetation cover of submerged plants. The flow model consists of the two-dimensional shallow water and continuity equations. The hydrodynamic equations are coupled firstly to the classic formulation for vegetation dynamics, and secondly to a modified version of the equation. The revised relationship for vegetation dynamics accounts for the influence of removal, transport, and resettlement of propagules on the growth rate of aquatic vegetation. Linear stability analysis of the eco-hydrodynamic problem is performed by enforcing the quasi-steady approximation. We obtain a dispersion relation disclosing the growth rate and the migration rate of the perturbations of vegetation density as a function of the wavenumber and the relevant flow and vegetation parameters. The present theory predicts the onset of vegetation patterns and includes an adequate wavelength selection mechanism. While uprooting initially reduces plant density, the analysis demonstrates that resettled propagules after removal are fundamental for further plant population increases and the development of vegetation patterns. The proposed framework is then validated against data available in the literature. Additionally, the presence of an upper threshold in terms of vegetation density, above which uniform vegetation cover is stable, might explain the absence of any spatial pattern and thus the extremely dense vegetation cover induced by climate change and invasive species in altered ecosystems.