We use stellar mass functions to study the properties and the significance of quenching through major galaxy mergers. In addition to SDSS DR7 and Galaxy Zoo 1 data, we use samples of visually selected major galaxy mergers and post merger galaxies. We determine the stellar mass functions of the stages that we would expect major merger quenched galaxies to pass through on their way from the blue cloud to the red sequence: 1: major merger, 2: post merger, 3: blue early type, 4: green early type and 5: red early type. Based on the similar mass function shapes we conclude that major mergers are likely to form an evolutionary sequence from star formation to quiescence via quenching. Relative to all blue galaxies, the major merger fraction increases as a function of stellar mass. Major merger quenching is inconsistent with the mass and environment quenching model. At z~0 major merger quenched galaxies are unlikely to constitute the majority of galaxies that transition the green valley. Furthermore, between z~0-0.5 major merger quenched galaxies account for 1-5% of all quenched galaxies at a given stellar mass. Major galaxy mergers are therefore not a significant quenching pathway, neither at z~0 nor within the last 5 Gyr. The majority of red galaxies must have been quenched through an alternative quenching mechanism which causes a slow blue to red evolution.