The Real Madrid forward announced on Sunday that he had exclusive custody of the baby boy, born in June to an anonymous American.
The news appeared on Real's website and Ronaldo's Twitter and Facebook pages.
"It is with great joy and emotion that I inform I have recently become a father to a baby boy," said Ronaldo.
The 25-year-old, who captained his country to the last 16 in South Africa before Portugal were knocked out by Spain, is not married.
"With the agreement of the mother, who wishes to remain anonymous, I will have exclusive custody of my son," added the world's most expensive footballer.
"I request everyone to fully respect my right to privacy [and that of the child] at least on issues as personal as these are."
Portuguese newspaper Correio da Manha reported on its website on Sunday that Ronaldo's mother, Dolores, and his sisters Elma and Katia, were dealing with the paperwork necessary to bring the child to Portugal.
It seems the news has come as light relief for the former Manchester United forward who said he felt like a "broken man" following his side's 1-0 second-round defeat by Iberian neighbours Spain.
The Spain defeat completed a disappointing tournament for Ronaldo, who failed to reproduce his scintillating club form for Portugal.