Select the curve object. Key Alt + C to open the Convert to menu and choose Mesh from Curve/Meta/Surf/Text or key 2. Key T to toggle open/close the Toolshelf. Check Keep Original in the Convert to dropdown. This adds a new mesh object, rather than converting the curve. This is useful because if you decide to change the curve later, you still can, without having to convert back to a curve, edit, and return to a mesh.
Select the object that you were loop cutting. Tab into Edit Mode. Key Ctrl + R to add a loopcut, position it where you like, LMB to confirm, and immediately RMB.
Key Ctrl + [comma] to change to Pivot point > Median point. Choose Transform Orientation > Normal. Enable Snapping and change the settings to Edge > Closest. Your 3D Viewport's bottom bar should now have these settings:
Drag on the green arrow until the loop cut snaps.