Is the statement: "I swear by Allah, if she is lying, then may Allah separate us" considered a supplication (du'a), and what is its ruling, and does it necessitate divorce, given the strong presumption of intending divorce, but not certainty?

The wording mentioned in the question is a type of supplication (du'a), and divorce does not occur merely by a supplicatory formula. Rather, it occurs with an explicit pronouncement of divorce or an equivocal expression of divorce accompanied by the intention of divorce. We recommend good companionship between the spouses and caution against any utterance that might negatively affect married life.