No. To fall apart is to collapse into pieces. Fallout is the airborne particles resulting from an explosion, most commonly attributed to radioactive particles after a nuclear explosion. It also metaphorically refers to the inevitable adverse consequences of an action, as in "the fallout from the team's failure to win may include the manager being sacked." However, that is not to be confused with "to fall out" which could means to become detached and drop away from, whether that is falling out of an aircraft, falling out of a race, or falling out of a military parade.