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.