How can one reconcile the prohibition of intimate contact with women during I'tikaf mentioned in Surat Al-Baqarah with the action of the Prophet, peace and blessings be upon him, of engaging in intimate contact while fasting, as mentioned in the Hadith of Aisha?

There is no contradiction between the verse and the hadith. The "intimate contact" (mubasharah) forbidden in the verse refers to sexual intercourse and anything less than that which leads to pleasure, in the context of the rules of i'tikaf (seclusion in the mosque). As for the "intimate contact" mentioned by Aisha (may Allah be pleased with her), it refers to the touching of skins, such as by hand or similar contact that does not involve the private parts. So, "intimate contact" in the verse refers to sexual intercourse, while "intimate contact" in the hadith refers to touching by hand. Sexual intercourse is forbidden in both cases. A fasting person should not engage in "intimate contact" such as kissing, etc., if they fear that their desire might be aroused.