Curtis left the MLS combine last month without explanation and did not attend the league's SuperDraft.
"Over the past four weeks, Curtis and the club tried to work through differing views," a club statement said. "Ultimately, both sides agreed to part ways."
Metro reported that assistant coachDenis Hamlett has been handling Curtis' duties in recent weeks.
Curtis joined the Red Bulls after the 2014 season and immediately replaced coachMike Petke with Jesse Marsch.
The Red Bulls enjoyed success during his tenure, winning the2015 Supporters' Shield for the best regular season record before also finishing first in the Eastern Conference in 2016. He also helped create the club's USL affiliate, Red Bulls II.
"I thank the New York Red Bulls for the opportunity," Curtis said in a statement released by the team. "It was a privilege and honor to serve the fans, players and staff. I am very proud of all the work and accomplishments of our team, both on and off the field, from the youth academy, through the USL team and to the first team. I would also like to thank the families of all of the staff and players as their support helped to fuel our achievements."