In a eulogy for Tutu, who died on 26 December, Mpumlwana suggested carrying forward his vision at a memorial service for the Anglican leader who helped lead his nation to reconciliation after being one of apartheid's most ardent foes.
Mpumlwana, a bishop of the Ethiopian Episcopal Church, who was in the struggle against apartheid with Tutu, made the comments during an interfaith memorial service.
"The idea of a rainbow nation was aspirational, something to strive for and to work towards. We pray for it to work for the promise of the post-apartheid South Africa.
"That has not been achieved, requiring the healing and reconciliation to address the woundedness of our society that makes us a violent society," said the South African church leader.
He spoke at one of the many services held in Tutu's honour throughout South Africa, where he is revered as he was lying in state at the St George's Cathedral in Cape Town and where his funeral service will be held on 1 January.
Mpumlwana rued the violent crime, corruption, gender-based violence, and senseless femicide against which Tutu railed.
"All this calls for a serious effort in an initiative for nation building towards Tutu's rainbow vision," said the SACC leader.
"Then there is the poverty and inequality that cries out for the transformation of the economic architecture of this country, to seriously address the inclusion of what we refer to as the excluded majority – Coloureds, Africans, people in the rural Bantustan areas and the small country towns for whom not much has changed," he said.
Among them are those who flee rural poverty for urban squalor in pursuit of elusive opportunities.
Mpumlwana made parallels between Tutu and Prophet Jeremiah, who said:
"If I say, 'I will not mention his word or speak any more in God's name,' his word is in my heart like a fire, a fire shut up in my bones. I am weary of holding it in; indeed, I cannot."
At the service, World Council of Churches (WCC) deputy general secretary Prof. Dr Isabel Apawo Phiri read a memorial message on behalf of WCS's acting general secretary Rev. Prof. Dr Ioan Sauca, lamenting Tutu's passing.
"Desmond Tutu had the uniquely vibrant personality, infectious laugh, and embrace of life that could disarm opposition and rally allies and followers. He was much loved and very loving. He was a churchman of deep faith and a committed ecumenist," said Phiri, a Malawian who studied in South Africa.
"But, as the Nobel Committee noted, he also brought to his work with the South African Council of Churches and the Anglican Church a steely clarity of purpose and an unwavering commitment to social justice for all South Africans."
She said that despite worldwide fame and praise for his successful work, he remained "a humble and earnest servant of God, intent on sharing the love of God for all God's people and the earth."
The Vatican's Nuncio in South Africa, Archbishop Peter Wells, also delivered a message.
"His Holiness Pope Francis was saddened to learn of the death of Archbishop Desmond Tutu, and he offers heartfelt condolences to his family and loved ones, mindful of his service to the Gospel to the promotion of racial equality and reconciliation in his native South Africa," said Wells.