The 38-year-old came to the final hole at the Thornblade Club in South Carolina needing a birdie to shoot 59.
After a well-struck iron finished behind the flag, Thatcher left his 15-foot birdie putt just short, as his ball cruelly broke from right to left across the front of the hole.
"Honestly, I just didn't think you could leave it short," he said. "I was just trying to pick the line. I'm actually ashamed of myself that I left it short for a chance at 59. That putt has been so quick over the last few years. I just thought if I got it on the right line, it would tumble down there.
"Halfway [to the hole], I thought I had done it, but it just went back up the rise there and fell right across the front edge."
Thatcher tapped in and settled for an 11-under 60, leaving him at 21 under for the championship and, at the time, just one stroke off the lead.
His bogey-free round included nine birdies and an eagle, with his final red number coming on the par-5 16th.
"All in all, obviously, you can't ever complain about that kind of round," he added.
Thatcher would have been the 14th player in golf history to record a sub-60 round and the sixth to do it on the Web.com Tour.
Russell Knox and Will Wilcox were the last two men to do it on that Tour, recording 59s two weeks apart in July 2013.
Al Geiberger was the first player to ever record a 59 in competitive play in 1977.