BigBlueFanGA is correct. Rivers flow due to elevation changes. The miniscule difference in the acceleration of gravity at the equator and at the poles is inconsequential. Any noncompressible fluid of a given density and specific gravity would also flow downhill. Simple hydrodynamics.
The New River is fairly close to me. It flows North from the mountains in North Carolina, "downhill" through Southwest VA and finally into West VA, where Bluestone Lake's discharge flows into the Ohio, then into the Mississippi, and then into the Gulf. Even when I was 9 years old, I understood why this river flowed North, while the other rivers my dad and I used to fish back then flowed South.