If you really want to use SphericalPlot3D, just switch off the surface plotting and control the 'mesh' lines to give a result that looks like a meridian line:
SphericalPlot3D[1, {theta, 0, 2 Pi} , {phi, 0, 2 Pi}, PlotStyle -> None, Mesh -> 1]
Or you could draw a circle in 3D space:
With[{radius = 1},
Graphics3D[
Line[Table[{radius Cos[a], 0, radius Sin[a]},
{a, 0, 2 Pi, Pi/120}]],
Axes -> True]]
