For the Buddha the "Middle Way" consisted of a spiritual practice that neither afflicted the body nor indulged its desires. The Buddha's concern was to free people from desire, the root of suffering, without handicapping the practitioner through asceticism.
For Aristotle the "Golden Mean" consisted in the correct, or virtuous, expression, of emotions: neither too much anger nor too little, neither too much pride nor too little, etc. Aristotle's concern was with excellence, or one could say, with moral beauty.
Sha'ul of Tarsus, St.Paul, also struggles to express a middle way in his letters. His middle way is between anti-nomianism and legalism. His concern is with people being reborn and conformed to the image of Jesus through the Holy Spirit. Who is Jesus? Jesus is the one totally surrendered to God, fully expressing the true image of humanity, and unconditionally loving toward others.
On the one hand he is concerned with releasing people from legalism- from judging oneself according to performance, ritual, and law- what came to be called halakhah. On the other hand he is concerned that the Church be virtuous and spiritually true and vital. But the engine that he wants is the response to Grace. The engine that he wants is trust, love, and open-ness to the sanctifying Spirit, which are the mechanisms of the New Covenant.
One could say that the Buddha is concerned with freedom from suffering, Aristotle is concerned with excellence, and Paul is concern is with relationships. His fundamental concern is with the relationship between the practitioner and God, between the practitioner and others, between the practitioner and him or herself. In each case the quality he is looking for is reconciliation, which is the transition to love.