Is it possible to be a whole person without relationships with others?
Probably not, if we take seriously the Bible's creation story. There we hear God saying, "It is not good for a person to be alone." More specifically, that is a reference to the creation of a woman to complement the man who is incomplete by himself. But it seems like it is bigger than just the man/woman complementary situation.
It appears like a principle that applies to everyone—namely, that with the presence of others in our lives, we are enabled to become more whole, healthy, and fulfilled. Conversely, isolating ourselves deprives us of spiritual sustenance that helps us to think better, breath better, live properly and ethically, and grow appropriately into what God intended.
Is this what Jesus was getting at when he said, "Where two or three are gathered in my name, there I am present with them"? That suggests that the Spirit of God works when people gather together prayerfully. Does it mean we need each other for the Spirit of God to come through most effectively? I think it does. Alone, we are deprived of the maximum power, wisdom, or love that flows from the heart of God.
It is spiritually dangerous to be a person of such power or independence that we become isolated and deprived of people who disagree, argue with, or refuse or affirm our plans and wishes. Terrible things have happened when dictators, kings, presidents, and other leaders are able to go their own way. Power that intimidates shuts off the necessary flow of God's loving and moderating Spirit coming from others. It is necessary for living rightly, and making sound and appropriate decisions to be meaningfully connected to others and trusting of others.