The musician goes on to say it was important for him and his wife to come back to social media to share how they're doing and how they plan to move forward, as well as to thank fans, medical staff and the country music community.
In the 27-minute video, Granger Smith says he is committed to River's legacy by looking for the good in "the worst situation we've ever been through."
The singer said the day River died, the family was outside in their Austin, Texas, home. He said he was outside, playing with 7-year-old daughter, London, as his sons, River and Lincoln Monarch, 5, were playing with water guns.
"I just had this thought, 'Soak up this moment because it's not going to last forever,'" he said. "Somewhere between 30 seconds and three minutes, we don't know, Amber and I are inside the pool gate, doing CPR on our son."
River was ultimately rushed to Dell Children’s Medical Center, where doctors and other medical staff worked to save him.
River died two days later.
"I think the platform that we immediately stand for is love those close to you, soak up those moments. Live for today. Live in the present because we're not guaranteed tomorrow.
"What happened to us defied all odds or all possibilities and we believe that, in our belief, that God gave us River for three years and that was his mission.
"I believe God takes anyone too soon," he said. "I'm not going to play around in mind that there's this fictitious timeline of River graduating from college or high school or playing football because I believe he was put on this Earth for that exact amount of time."
The last 20 minutes of the video included photos and clips of River's life, which was also shown at his funeral service.
The Smiths have asked fans to donate in River's name to Dell Children's Medical Center. All proceeds from a River Smith tribute shirt on his website will go toward the hospital in River's honor. More than $130,000 has been raised.
Smith said he will continue to make music and tour with his family.
"We'll find the meaning and the good in this ... Don't feel sorry for us. We feel very blessed," Smith said. "We had an incredible boy for three years, and we feel really good about that."