Only two other men's and women's quarterfinals were completed before all play was called off on a wet day that also included three-time major champion Andy Murray's withdrawal.
The No. 1-seeded Zverev, who beat his older brother, Mischa, in the third round, also eliminated 2015 Washington champion Nishikori at the hard-court tuneup for the U.S. Open a year ago en route to the title.
Germany's Zverev, 21, improved to 14-2 in his four appearances at the Citi Open.
Tsitsipas is coming off his first run to the fourth round at a Grand Slam tournament, last month at Wimbledon.
On the other half of the draw, Murray cited fatigue when he withdrew from his quarterfinal match Friday, hours before he was supposed to face 19-year-old Alex de Minaur, who was given a walkover into the semifinals.
Storms earlier in the week forced some men to play twice in one day.
The Associated Press contributed to this report.