Newtonian gravitational potential sourced by a homogeneous circular ring in arbitrary dimensional Euclidean space takes a simple form if the spatial dimension is even. In contrast, if the spatial dimension is odd, it is given in a form that includes complete elliptic integrals. In this paper, we analyze the dynamics of a freely falling massive particle in its Newtonian potential. Focusing on circular orbits on the symmetric plane where the ring is placed, we observe that they are unstable in 4D space and above, while they are stable in 3D space. The sequence of stable circular orbits disappears at $1.6095cdots$ times the radius of the ring, which corresponds to the innermost stable circular orbit (ISCO). On the axis of symmetry of the ring, there are no circular orbits in 3D space but more than in 4D space. In particular, the circular orbits are stable between the ISCO and infinity in 4D space and between the ISCO and the outermost stable circular orbit in 5D space. There exist no stable circular orbits in 6D space and above.
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