Nadal, who was on court for four hours and 48 minutes in a tie which finished at 2.04am local time in NY (7.04am BST), commented on the humidity of the playing conditions that has troubled so many players in this year's Open, and broke down his performance.
"I'm very sorry for Dominic", Nadal said.
Rafael also ensured he gave some words of encouragement to Dominic and admitted he said "sorry" to his opponent. "It means a lot to me". There are the matches he actually loses - as the past and present world No. 1, a 17-time major victor, these are rare - but not before leaving behind a Mediterranean Sea's worth of sweat and tears. A tough match against a great opponent.
"The conditions have been tough out there, too. But in few days I will look back and will remember how great it was to play in front of a packed Arthur Ashe - this great match". He played a good match.
"He is a very well educated guy and a top person".
"I just wish him all the very very best".
"I think last three sets I improved". I'm happy, of course, to be in the semifinals again. "I'm playing a lot of hours this year at this lovely stadium".
Nadal won just seven points for the entire set, his nine unforced errors handing the set to Thiem on a plate.
The Spaniard was able to bounce back and win the next two sets 6-4, 7-5, but not without a fight from Thiem.
After Nadal roared back to take a 2-1 lead, the fourth set went to a tiebreak.
Nadal, the world No 1 from Spain, put a terrible start behind him - he won just seven points in the first set - to overcome the formidable challenge of Thiem, the world No 9 from Austria, 0-6, 6-4, 7-5, 6-7 (4-7), 7-6 (7-5). Nadal said he told Thiem not to despair, seeing the potential for greatness in his future.
There are no chill Rafael Nadal matches, like oh cool, I'll watch some tennis and my blood pressure will stay level. "I've played many, many hours on this wonderful court".
While Nadal was tight-lipped about his match strategy, Thiem was bit more forthcoming on what he has to do to win.
Asked about that at his news conference, Thiem said with a chuckle: "Well, I don't think he's really sorry".
"He's a great guy". They have more value, more personal value, when you come back from tough moments, no? I think we nearly all the time have great matches. That's one more today.
Nadal will next face 2009 champion Juan Martin del Potro.
Williams was on track to meet defending champion Sloane Stephens in the semi-finals but her third-seeded countrywoman was dumped out of the tournament via a 6-2, 6-3 loss to 19th seed Anastasija Sevastova.
While that encounter was settled in straight sets, Thiem was agonisingly close to his first major victory over Nadal on this occasion, and seeing the opportunity slip from his grasp is something that he says will never leave him.