The fraction of ionizing photons that escape high-redshift galaxies sensitively determines whether galaxies reionized the early Universe. However, this escape fraction cannot be measured from high-redshift galaxies because the opacity of the intergalactic medium is large at high redshifts. Without methods to measure the escape fraction of high-redshift galaxies indirectly, it is unlikely that we will know what reionized the Universe. Here, we analyze the far-ultraviolet (UV) H I (Lyman series) and low-ionization metal absorption lines of nine low-redshift, confirmed Lyman continuum emitting galaxies. We use the H I covering fractions, column densities, and dust attenuations measured in a companion paper to predict the escape fraction of ionizing photons. We find good agreement between the predicted and observed Lyman continuum escape fractions (within 1.4σ) using both the H I and ISM absorption lines. The ionizing photons escape through holes in the H I, but we show that dust attenuation reduces the fraction of photons that escape galaxies. This means that the average high-redshift galaxy likely emits more ionizing photons than low-redshift galaxies. Two other indirect methods accurately predict the escape fractions: the Lyα escape fraction and the optical O III/O II flux ratio. We use these indirect methods to predict the escape fraction of a sample of 21 galaxies with rest-frame UV spectra but without Lyman continuum observations. Many of these galaxies have low escape fractions (fesc ≤ 1%), but 11 have escape fractions >1%. Future studies will use these methods to measure the escape fractions of high-redshift galaxies, enabling upcoming telescopes to determine whether star-forming galaxies reionized the early Universe.