Motivated by an anomaly in $R(D^{(*)})={rm BR}(bar{B}rightarrow D^{(*)} tau^-bar{ u})/{rm BR}(bar{B}rightarrow D^{(*)} l^-bar{ u})$ reported by BaBar, Belle and LHCb, we study $R(D^{(*)})$ in a general two Higgs doublet model (2HDM). Although it has been suggested that it is difficult for the 2HDM to explain the current world average for $R(D^{(*)})$, it would be important to clarify how large deviations from the standard model predictions for $R(D^{(*)})$ are possible in the 2HDM. We investigate possible corrections to $R(D^{(*)})$ in the 2HDM by taking into account various flavor physics constraints (such as $B_c^-rightarrow tau^- bar{ u}$, $brightarrow sgamma$, $brightarrow s l^+l^-$, $Delta m_{B_{d,s}}$, $B_srightarrow mu^+mu^-$ and $tau^+tau^-$, and $B^-rightarrow tau^- bar{ u}$), and find that it would be possible (impossible) to accommodate the 1$sigma$ region suggested by the Belles result when we adopt a constraint ${rm BR}(B_c^-rightarrow tau^- bar{ u})le30~%$ (${rm BR}(B_c^-rightarrow tau^- bar{ u})le10~%$). We also study productions and decays of heavy neutral and charged Higgs bosons at the Large Hadron Collider (LHC) experiment and discuss the constraints and implications at the LHC. We show that in addition to well-studied production modes $bgrightarrow tH^-$ and $ggrightarrow H/A$, exotic productions of heavy Higgs bosons such as $cgrightarrow bH^+,t+H/A$ and $cbar{b}rightarrow H^+$ would be significantly large, and the search for their exotic decay modes such as $H/Arightarrow tbar{c}+cbar{t},~mu^pmtau^mp$ and $H^+rightarrow cbar{b}$ as well as $H/Arightarrow tau^+tau^-$ and $H^+rightarrow tau^+ u$ would be important to probe the interesting parameter regions for $R(D^{(*)})$.
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