The radiative decay of $K$-shell vacancies of argon, krypton, and xenon, and $L$-shell vacancies of krypton and xenon, has been observed with a Si(Li) x-ray detector for thin gas targets of these atoms excited by 1.5- to 5.0-MeV protons. X-ray yields were measured as a function of target thickness under single-collision conditions, and inner-shell ionization cross sections were determined using atomic fluorescence yields. The $L$-shell ionization cross sections obtained are in good agreement with theoretical calculations in the plane-wave Born approximation. The $K$-shell cross sections show good agreement with these calculations for the argon $K$ shell, but fall above the theoretical values for krypton and xenon, the discrepancy with theory increasing with increasing atomic number. The binary-encounter model was also compared to the experimental $K$-shell ionization cross sections, and for all gases the agreement is somewhat worse. $\frac{K\ensuremath{\beta}}{K\ensuremath{\alpha}}$ relative intensities for krypton and xenon were determined, and good agreement of these values with the results of experiments using other modes of excitation is shown.