Future radial velocity, astrometric, and direct-imaging surveys will find nearby Earth-sized planets within the habitable zone in the near future. How can we search for water and oxygen in those nontransiting planets? We show that a combination of high-dispersion spectroscopic and coronagraphic techniques is a promising technique to detect molecular lines imprinted in the scattered light of Earth-like planets (ELPs). In this method, the planetary signals are spectroscopically separated from telluric absorption by using the Doppler shift. Assuming a long observing campaign (a 10-day exposure) using a high-dispersion spectrometer (R=50,000) with speckle suppression on a 30-m telescope, we simulate the spectra from ELPs around M dwarfs (whose stellar effective temperature is 2750-3750 K) at 5 pc. Performing a cross-correlation analysis with the spectral template of the molecular lines, we find that raw contrasts of $10^{-4}$ and $10^{-5}$ (using Y, J, and H bands) are required to detect water vapor at the 3 $sigma$ and 16 $sigma$ levels, respectively, for $T_star$=3000 K. The raw contrast of $10^{-5}$ is required for a 6 $sigma$ detection of the oxygen 1.27 $mu$m band. We also examine possible systematics, incomplete speckle subtraction, and the correction for telluric lines. When those are not perfect, a telluric water signal appears in the cross-correlation function. However, we find the planetary signal is separated from that resulting from the velocity difference. We also find that the intrinsic water lines in the Phoenix spectra are too weak to affect the results for water detection. We conclude that a combination of high-dispersion spectroscopy and high-contrast instruments can be a powerful means to characterize ELPs in the extremely large telescope era.
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