The Portugal international, 37, and Rodriguez, 28, were expecting twins.
Their baby girl survived, and they said her birth “gives us the strength to live this moment with some hope and happiness”.
“Our baby boy, you are our angel. We will always love you,” they wrote.
— Cristiano Ronaldo (@Cristiano) April 18, 2022
In a statement on social media, they said: “It is with our deepest sadness we have to announce that our baby boy has passed away.
“We are all devastated at this loss and and we kindly ask for privacy at this very difficult time.”
The pair had announced the pregnancy in October.
United tweeted: “Your pain is our pain. Sending love and strength to you and the family at this time.”
Teammate Marcus Rashford tweeted: “Thoughts are with you and Georgina brother I’m so sorry.”
Ronaldo’s former side Real Madrid said the club, its president and board of directors “are deeply saddened” adding: “Real shares in the family’s pain and wishes to send them all of our warmth.”
Manchester City, Leeds United and the Premier League have also tweeted messages of condolence.
Ronaldo has a son Cristiano Jr, who was born in 2010 and twins Eva and Mateo who were born in 2017. He also has a daughter with Rodriguez – Alana Martina, who was born in 2017.