Am I obligated to arrange my father's marriage after he divorced his wife based on a rumor and incitement to divorce, and after he promised me he would divorce her and marry him to another woman?

What your uncle did by accusing your stepmother is unlawful. Either she is innocent, in which case he has lied and slandered her honor, or she is guilty, in which case he should have concealed her fault and not exposed her. You also erred by relaying the information without verification, and you should have concealed her fault if the information were true. As for your father's dream, no legal ruling is based upon it, and it cannot be a reason for a disliked divorce unless there is a [valid] reason. You are not obligated to seek a marriage for your father after his wife's divorce, but you will be rewarded for helping him in that regard.