The International Tennis Federation (ITF) and investment group Kosmos Tennis made the announcement following a three-month review of the logistical and regulatory challenges that have arisen since the start of the pandemic.
The 2020 event was originally scheduled for April but was postponed because of the pandemic.
They remain at the László Papp Arena in Hungarian capital Budapest, set to be played on an indoor clay court.
But various exhibitions have been taking place, including a series organized by Novak Djokovic that had people in the stands and zero social distancing - and the No. 1-ranked Serb was one of four players who tested positive for the coronavirus.
Haggerty said: "Earlier this year, we took the hard but necessary decision to postpone the first edition of the new-look Fed Cup in Budapest due to the COVID-19 pandemic".
"While it is regrettable that the Finals will not be able to take place this year, we are confident that we will be able to host a superb event in Budapest in April 2021", said Haggerty.
The Davis Cup men's competition was supposed to have qualifying matches for future finals this September and the 2020 finals in November. "The ITF and Kosmos Tennis look forward to delivering an outstanding competition in 2021, when it is safe and feasible to do so", said Gerard Pique, Kosmos Tennis President.
The worldwide tennis body took the decision days after four leading tennis players, including world no. 1 Novak Djokovic, tested positive for coronavirus. Rafa Nadal led Spain to the title past year, beating Canada in the final.