The Florida-based Scot, ranked 20th in the world and 10th on the World Points list for next month's event, had hoped to be one of three wildcard picks.
But Clarke chose Lee Westwood, Martin Kaymer and the in-form Thomas Pieters.
"It's probably as difficult a phone call as I've ever had to make in golfing terms," Clarke said.
"It was very difficult and I missed out myself on a captain's pick in 2008 when I won two weeks before the team was picked.
"So, making the call I knew how bad he was going to be feeling. Consequently, I felt bad having to do it.
"But that's part of the captain's remit and I must say he was very gracious and he understood where I was coming from."
While the experienced Westwood, who will make his 10th appearance in the biennial clash at Hazeltine, Minnesota, and Kaymer were expected to be two of Clarke's choices, Knox, 31, lost out to 24-year-old Pieters.
The Belgian won the Made in Denmark title at the weekend, after a fourth-place finish at the Rio Olympics and a  runner-up finish at the Czech Masters.
It means the 2016 event will be only the third since the Second World War that no Scot has featured in the European team, following Nick Faldo's defeated 2008 team and Colin Montgomerie's victorious group in 2010.
Knox, just one place away from being an automatic pick, would have qualified if he had been a European Tour member when he won the WGC-HSBC Champions last November and boosted his case by adding the Travelers Championship earlier this month.
"Last week I played with Thomas Pieters and Thomas knew why I was playing with him and played unbelievable and shot 62 and, on top of that, won the tournament," Northern Irishman Clarke added.
"In Lee Westwood I have one of most experienced Ryder Cuppers of all time - it's his 10th appearance.
"Martin Kaymer is somewhat of a Ryder Cup stalwart and another man I have enormous trust in."
Telling Russell Knox he had missed out on a captain's pick for Europe's Ryder Cup team was the hardest call captain Darren Clarke "has ever had to make".