We present a search for a standard model Higgs boson decaying to two W bosons that decay to leptons using the full data set collected with the CDF II detector in s√=1.96 TeV pp¯ collisions at the Fermilab Tevatron, corresponding to an integrated luminosity of 9.7 fb−1. We obtain no evidence for production of a standard model Higgs boson with mass between 110 and 200 GeV/c2, and place upper limits on the production cross section within this range. We exclude standard model Higgs boson production at the 95% confidence level in the mass range between 149 and 172 GeV/c2, while expecting to exclude, in the absence of signal, the range between 155 and 175 GeV/c2. We also interpret the search in terms of standard model Higgs boson production in the presence of a fourth generation of fermions and within the context of a fermiophobic Higgs boson model. For the specific case of a standard-model-like Higgs boson in the presence of fourth-generation fermions, we exclude at the 95% confidence level Higgs boson production in the mass range between 124 and 200 GeV/c2, while expecting to exclude, in the absence of signal, the range between 124 and 221 GeV/c2.