We study the role of disorder in the exciton spectra in two-dimensional (2D) semiconductors. These can be heterostructures, thin films and multilayers (so-called van der Waals structures) of organometallic perovskites, transition metal dichalcogenides and other semiconductors for optoelectronic applications. We model the disorder by introduction of a fractional Laplacian (with Lévy index μ, defining the degree of disorder) to the Scrödinger equation with 2D Coulomb potential. Combining analytical and numerical methods, we observe that the exciton exists only for μ > 1, while the point μ = 1 (strongest disorder) corresponds to the exciton collapse. We show also that in the fractional (disordered, corresponding to 1 < μ < 2; μ = 2 corresponds to the ordered case) 2D hydrogenic problem, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number n but also on orbital m. These features can have a profound influence on the lifetime of optically generated excitons in the above 2D semiconductor structures. They should be taken into account while designing the photovoltaic cells, nanolasers and optical spintronics devices, where 2D excitons play a significant role.