Is it permissible to greet an uncle who touched sensitive parts of his niece's body when she was young, and is ghusl (ritual purification bath) required as a result of that?

There is no doubt that enjoying forbidden acts is among the most heinous sins. However, if your uncle abandons those prohibited actions and shows signs of repentance and righteousness, then there is no harm in shaking his hand and treating him as a mahram. Do not be excessive in mingling with him or being alone with him until his righteousness and complete chastity become clear to you. And know that performing ghusl after shaking his hand is a mistaken approach that opens a door for Satan to cause you to fall into whisperings (waswasah). So, beware of that and turn away from these whisperings. Among what helps in getting rid of these whisperings is seeking Allah's help, sincerely turning to Him, and supplicating frequently.