The 31-year-old British star, who returned in June from an 11-month layoff with a right hip injury, also withdrew from next week's ATP Toronto Masters, citing the fatigue from three grueling three-set Washington victories.
"I also need to be careful and listen to my body as I come back from a long-term injury".
"I don't think I should be put in a position like that", Murray said after the victory. It's not good for the players, not good, I don't think, for anyone involved in the event.
The Brit was due to play young Aussie Alex De Minaur in a quarter-final on Friday but Murray decided against competing due to exhaustion.
The extraordinarily late finish to his match with Copil gave him little time to recover for his quarter-final and in the aftermath he voiced his annoyance at the scheduling, hinting that he may withdraw to preserve his exhausted body. On Monday, he didn't play his opening-round match until 10pm because of a five-hour rain delay.
Murray is playing just his third event of 2018 after hip surgery in January. The 19-year-old is projected to soar past his previous career-best world ranking of 68 and could make the top 50.
Tournament Director Keely O'Brien added: "I am so grateful that Andy, an incredible champion, came back to D.C.to begin what we all know will be a great comeback. And I'm disappointed with that, because I know that the weather's tricky and I know it is for the scheduling, but it's a very hard position to be in". Zverev joked after advancing, "I think I'm the only one that's allowed to buy a drink in the USA, right?" "Appreciate all the support this week in Washington, it's unbelievable to be back playing". "And I'm disappointed with that, because I know that the weather's tricky and I know it is for the scheduling, but it's a very hard position to be in".
Murray, now ranked at No. 832 in the world, is a three-time Grand Slam champion and back-to-back Olympics champion.
The frustrations included Murray leading the first-set tie-breaker 5-0 only to drop the next seven points and the set.
When he got to the sideline after the match concluded, he covered his face with a towel and cried, his chest heaving.
Defending champion and world number three Alexander Zverev of Germany was set to face Japanese seventh seed Kei Nishikori, with the victor to face Tsitsipas, and local hero Denis Kudla was to meet Russian 16th seed Andrey Rublev in other quarter-finals.