The Estadio Municipal de Balaidos was declared unsafe by Vigo's city council, its owner, on Saturday.
Abel Caballero, the city's mayor, had earlier told Spanish media it would be impossible for the game to go ahead as planned. No new date has yet been set.
Real are top of La Liga, one point ahead of second-placed Barcelona with two matches in hand.
Luis Enrique's side beat Athletic Bilbao 3-0 on Saturday, with Paco Alcacer scoring his first league goal since his move from Valencia last August.
The Galicia region of north-west Spain, of which Vigo is a part, has been hit by winds of up to 85mph in recent days, causing the postponement of Friday's La Liga match between Deportivo La Coruna and Real Betis.
However, the decision to call off Celta Vigo's match has upset Alaves, their Copa del Rey semi-final opponents.
Last season's Segunda Division champions have released a statement asking for their semi-final second leg against Celta on Wednesday to be postponed. The clubs drew the first game 0-0.
They claim Celta will have an unfair advantage with the extra rest, with Alaves away to Sporting Gijon on Sunday.
This match has been postponed to play at a later date.
Real Madrid's La Liga game at Celta Vigo on Sunday has been postponed after storms damaged the hosts' ground.