The STAR collaboration reports a measurement of the transverse single-spin asymmetries, A(N), for neutral pions produced in polarized proton collisions with protons (pp), with aluminum nuclei (pAl) and with gold nuclei (pAu) at a nucleon-nucleon center-of-mass energy of 200 GeV. Neutral pions are observed in the forward direction relative to the transversely polarized proton beam, in the pseudorapidity region 2.7 < eta < 3.8. Results are presented for pi(0)s observed in the STAR forward meson spectrometer electromagnetic calorimeter in narrow Feynman x (x(F)) and transverse momentum (p(T)) bins, spanning the range 0.17 < x(F) < 0.81 and 1.7 < p(T) < 6.0 GeV/c. For fixed x(F) < 0.47, the asymmetries are found to rise with increasing transverse momentum. For larger x(F), the asymmetry flattens or falls as p(T) increases. Parametrizing the ratio r(A) = A(N)(pA)/A(N)(pp) = A(P) over the kinematic range, the ratio r(A) is found to depend only weakly on A, with < P > = -0.027 +/- 0.005. No significant difference in P is observed between the low-p(T) region, p(T) < 2.5 GeV/c, where gluon saturation effects may play a role, and the high-p(T) region, p(T) > 2.5 GeV/c. It is further observed that the value of A(N) is significantly larger for events with a large-p(T) isolated pi(0) than for events with a nonisolated pi(0) accompanied by additional jetlike fragments. The nuclear dependence r(A) is similar for isolated and nonisolated pi(0) events.