We present an analysis of the interstellar medium (ISM) toward the gamma -ray supernova remnant (SNR) W44. We used NANTEN2 super(12)CO(J = 2-1) and super(l2)CO(J = 1-0) data and Arecibo H I data in order to identify the molecular and atomic gas in the SNR. We confirmed that the molecular gas is located in the SNR shell with a primary peak toward the eastern edge of the shell. We newly identified high-excitation molecular gas along the eastern shell of the SNR in addition to the high-excitation broad gas previously observed inside the shell; the line intensity ratio between the super(12)CO(J = 2-1) and super(l2)CO(J = 1-0) transitions in these regions is greater than ~1.0, suggesting a kinetic temperature of 30 K or higher, which is most likely due to heating by shock interaction. By comparing the ISM with gamma -rays, we find that target protons of hadronic origin are dominated by molecular protons of average density around 200 cm super(-3), where the possible contribution of atomic protons is 10% or less. This average density is consistent with the recent discovery of the low-energy gamma -rays suppressed in 50 MeV-10 GeV as observed with AGILE and Fermi. The gamma -ray spectrum differs from place to place in the SNR, suggesting that the cosmic-ray (CR) proton spectrum significantly changes within the middle-aged SNR perhaps due to the energy-dependent escape of CR protons from the acceleration site. We finally derive a total CR proton energy of ~10 super(49) erg, consistent with the SN origin of the majority of the CRs in the Galaxy.