Defender Ramos, one of the club captains and a world champion with Spain last year in South Africa, had the Cup on top of his head as the bus inched past thousands of Real fans celebrating in the centre of the Spanish capital.
As he lowered the imposing silver trophy, which weighs nine kilograms and measures 78 centimetres, it slipped from his grasp and under the front of the open-top vehicle.
The bus continued for several metres before the driver realised and photographs reproduced in local media suggested it had suffered serious damage.
This was later confirmed by Federico Alegre, the curator of the club's trophy collection, who told sports daily Marca Real had had to put a replica of the Cup on display at their Bernabeu stadium on Thursday.
"There's always a plan B," Marca quoted him as saying. "I am not going to get into the state of the Cup because everyone saw from where it fell and that a bus ran over it ... you can imagine the state it was in," he added.
"We'll have to see if we can sort it out. The patient is in a coma but there are comas that can be overcome."
Cristiano Ronaldo's 103rd-minute header sealed Real's victory over arch rivals Barcelona in Valencia, their first domestic Cup win since 1993 and their first trophy under coach Jose Mourinho.
Watch a video of Ramos dropping the trophy off the bus during Real Madrid's victory parade below...