I think you're asking if Stein's result extends to $R^2$ for $p>2$. This was proved in
J. Bourgain, MR 874045 Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69--85.
See
W. Schlag, MR 1388870 A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), no. 1, 103--122.
for a discussion and more precise results.
