The 31-year-old has not played since the WGC-FedEx St Jude Invitational after withdrawing from The Northern Trust with an ankle injury.
Reed is currently ninth in the Ryder Cup standings and will not earn enough points to lift himself into the top six and qualify.
The American, who has played a key role at the biennial showpiece in recent years, could still be one of Steve Stricker's six captain's picks for Team USA, although Reed faces stiff competition from the likes of Webb Simpson, Jordan Spieth, Patrick Cantlay, Harris English, Tony Finau and Daniel Berger.
He said in a statement on the PGA Tour website: "The good news is, my ankle is OK. The bad news is I've been in the hospital with bilateral pneumonia.
"I'm on the road to recovery. Once I'm cleared from the doctors I look forward to returning. I wish you all the best and I can't wait to get back out there!
"Thank you so much for your support, it means a lot to me. Also, a special thank you to the doctors, nurses and staff at The Methodist Hospital in the Texas Medical Center."
Reed, dubbed “Captain America” due in part to his Ryder Cup performances, has nine PGA Tour wins to his name and last won in January at the Farmers Insurance Open. He claimed his maiden major at the 2018 Masters by one stroke over Rickie Fowler.
Meanwhile, Tony Finau took his second PGA Tour title with a playoff win over Cameron Smith at The Northern Trust.
The final round of the tournament in New Jersey was pushed into Monday after Hurricane Henri headed towards the north-eastern US coast and the result was delayed as Finau and Smith both finished on 20-under.
Australian Smith twice made back-to-back birdies on the back nine to recover after a double bogey on the fifth to card a four-under 67, while Finau went birdie-eagle-birdie on holes 12-14 to help himself to a six-under 65 to grab a share of the lead after 72 holes.
Smith's tee shot on the first playoff hole - the 18th - landed out of bounds, and Finau carded a par four to take the title, with overnight leader Jon Rahm also overtaken after shooting a 69 to finish third.
Finau said: “This one took everything I had because I knew I had to get to 20 under. I was chasing down the best player in the world. Jon is a good friend but I know what type of game he has, and he's not going to let up on a lead.
“So I knew I was going to have to catch him eventually and pass him, and Cam was playing well. I gave everything I had. I hit a great putt on 17 I thought was to go in, it would have been nice to have a little cushion on the 18th tee.”
Alex Noren, Tom Hoge and Justin Thomas tied for fourth on 15 under while Ireland's Shane Lowry was tied for 11th on 12-under for the tournament.
Lee Westwood was tied for 27th on nine under, a shot ahead of Ian Poulter and two ahead of Rory McIlroy.