The Manchester United forward, 37, said that their “baby boy had passed away” in a post on Instagram on Easter Monday.
Ronaldo and Ms Rodriguez announced they had been expecting twins at the end of October 2021.
The footballer said in a statement: “It is with great sadness that we have to announce our baby boy has passed away. It is the greatest pain that any parent can feel.
“Only the birth of our baby girl gives us strength to live this this moment with some hope and happiness.
“We would like to thank the doctors and nurses for all their expert care and support. We are all devastated at this loss and we kindly ask for privacy at this very difficult time.
“Our baby boy, you are our angel. We will always love you.”
Ronaldo and Ms Rodriguez have been dating since 2016 and first met at a Gucci store in Madrid where she worked as a sales assistant.
Responding to the news, teammate Marcus Rashford wrote: “Thoughts are with you and Georgina brother. I’m so sorry.”
Match of the Day host and former England striker Gary Lineker said: “Terribly sad news. Sincere condolences to you and your family.”
Ronaldo and Ms Rodríguez share their daughter Alana together while Ronaldo also has son Cristiano Jnr and twins Eva and Mateo.
In 2015, the footballer said becoming a father had been an “incredible” experience.
“It’s been incredible. I never imagined that my life would change so much,” he told Spanish news agency EFE at the time.
“It was always my dream to have a child young, when I was 25. He’s changed the way I think, he’s always with me, supporting me.”