He said white people "escaped" institutions once they were opened to all races.
He used as an example four Catholic schools in the Pretoria city centre that had only white pupils in the 1960s but now had hardly a single white pupil.
"They have all withdrawn from the schools. So white people have a problem," he said.
Slattery said the fact that the country was grappling with racial tension more than two decades into democracy was a huge frustration for people yearning for a nonracial society.
White people were impoverishing themselves in the process by not being sensitive to other people's cultures, he said.
Slattery said the answer was patience, saying that more than a century after the Anglo-Boer War English and Afrikaners found it difficult to mix and socialise.
"It takes a long time. It is a question of trust," he said, adding that white people needed to learn to trust their brothers and sisters.
Malusi Mpumlwana, a Bishop of the Ethiopian Episcopal Church, shared the same sentiments, saying all churches in Pretoria had gone black because whites "have escaped".