We report on charmonium measurements J/psi (1S), psi' (2S), and chi(c) (1P) in p + p collisions at root s = 200 GeV. We find that the fraction of J/psi coming from the feed-down decay of psi' and chi(c) in the midrapidity region (vertical bar y vertical bar < 0: 35) is 9.6 +/- 2.4% and 32 +/- 9%, respectively. We also present the p(T) and rapidity dependencies of the J/psi yield measured via dielectron decay at midrapidity (vertical bar y vertical bar < 0.35) and via dimuon decay at forward rapidity (1.2 < vertical bar y vertical bar < 2.2). The statistical precision greatly exceeds that reported in our previous publication Phys. Rev. Lett. 98, 232002 (2007). The new results are compared with other experiments and discussed in the context of current charmonium production models.