"I have very sad news. My two and a half year year old son Henry has passed away," Delaney wrote.
According to the actor, who starred in the Amazon comedy series for three seasons, his son was diagnosed with a brain tumor in 2016, shortly after his first birthday. He received surgery and treatment through the early part of 2017, but the cancer returned late last year. Henry died in January.
"My wife and Henry's older brothers and I are devastated of course," he wrote. "Henry was a joy. He was smart, funny, and mischievous and we had so many wonderful adventures together, particularly after he'd moved home following fifteen months living in hospitals."
According to Delaney, the tumor and surgeries had left Henry with "significant physical disabilities," but he had learned how to communicate through sign language.
"His drive to live and to love and to connect was profound," Delaney said.
Delaney praised the doctors and nurses of the National Health Service, the UK's publicly funded national healthcare system, who he said "will be my heroes until the day I die."
"I am desperately sad right now, but I can say with authority that there is good in this world," he wrote.
Delaney asked that donations be directed to Rainbow Trust or Noah's Ark, two organizations that provide support to families with sick children.
"I am astonished by the love-in-action displayed by Henry's mom and his brothers," Delaney added. "They are why I will endeavor to not go mad with grief. I don't want to miss out on their beautiful lives."