An inclusive search for the standard model Higgs boson using the four-lepton final state in proton-antiproton collisions produced by the Tevatron at sqrt(s) = 1.96 TeV is conducted. The data are recorded by the CDF II detector and correspond to an integrated luminosity of 9.7 /fb. Three distinct Higgs decay modes, namely ZZ, WW, and tau-tau, are simultaneously probed. Nine potential signal events are selected and found to be consistent with the background expectation. We set a 95% credibility limit on the production cross section times the branching ratio and subsequent decay to the four lepton final state for hypothetical Higgs boson masses between 120 GeV/c^2 and 300 GeV/c^2.