The phrase "kept one eye on" essentially means to be vigilant or watchful. Some of its antonyms include terms like careless, inattentive, negligent, unconcerned, and disinterested among others. When one does not keep an eye on something, they can be regarded as neglectful, unobservant or careless. The opposite of watching over something attentively, therefore, is to ignore it completely. Other antonyms for the phrase may include ignorant, uninterested, or indifferent, depending on the context. In all, the antonyms for "kept one eye on" are terms that denote a lack of attention or concern.