Fish, ranked ninth, is pulling out because of a health scare, the U.S. Tennis Association said on its website.
Hours after losing his most recent match, Fish, the highest-ranked American player, woke up in the middle of the night feeling ill and immediately went to be checked by a doctor. He saw another doctor the next day, too, and the diagnosis was fatigue.
The USTA said Fish is expected to recover fully.
"We don't want to get exactly into what the symptoms were. He didn't feel normal. He said, 'Hey, something's wrong here.' He was concerned about it," Fish's agent, John Tobias, said in a telephone interview with The Associated Press on Monday. "His body got out of whack, caused by a number of different variables ... (but) they ruled out any potential long-term issues."
Late on Monday, Fish wrote on his Twitter page: "Thanks all for the nice messages. Resting comfortably with family. Hope to be back out there soon."
Ryan Harrison is replacing Fish on the U.S. roster.
Monfils will sit out because of an abdominal injury. He started experiencing pain during practice Sunday and an MRI test showed a lesion on his left oblique muscle. He will need to rest for 15 days.
The French tennis federation said Monday that coach Guy Forget will soon name a replacement for the 14th-ranked Monfils.
The quarterfinal matches, at Monte Carlo Country Club in Roquebrune, France, start Friday. The winner will face the winner of the quarterfinal between Spain and Austria.
Fish lost 6-1, 6-3 on Thursday to Juan Monaco in the quarterfinals of the Sony Ericsson Open at Key Biscayne, Fla. The 30-year-old Fish appeared to be fine physically and said afterward he would have preferred more taxing conditions for the afternoon match.
"It was like the weather was almost too good," said Fish, who lost about 30 pounds after having knee surgery in September 2009 and prides himself on his fitness. "It wasn't that humid and it was a nice, comfortable temperature. So I was expecting it to be hotter. I wanted it to be kind of a Florida day. It was an L.A. day, I think."
From Switzerland, Fish went home to California, before traveling to Marseille, then Dubai, then home again, before tournaments at Indian Wells and Key Biscayne.
"Just a crazy amount of travel, and the time-zone changes, kind of wears you out after a while," Tobias said.
On Thursday night in Miami, Fish didn't feel well and saw a doctor. His parents live a few hours away in Florida, and he went to see the family doctor on Friday. After spending the weekend at his parents' home, Fish was going to fly to his place in California on Monday, Tobias said.
The agent said Fish will consider making changes to his schedule now, but he could still play next week at Houston. And Fish intends to compete on the European clay-court circuit at Rome and Madrid before the next Grand Slam tournament, the French Open, which begins May 27. Tobias said that "absolutely, 100 percent" Fish will play at Roland Garros.
"We're just going to scale him back a little bit and make sure that he arrives at a tournament well-rested and ready to compete," Tobias said. "His commitment to Davis Cup is like none other. It's absolutely killing him to miss it this time."
Information from The Associated Press was used in this report.