Police are investigating the incident, but believe that Mr Delves fell accidentally 32ft from a balcony in the early hours of yesterday morning.
Neighbours of Mr Delves have spoken of the community's deep shock and sadness.
John Bollard (56) told the Herald: "The whole area is reeling from this terrible tragedy. Robert was a quiet and gentle man who was a wonderful neighbour and a great husband and father.
"This is such a terrible tragedy to hit a lovely family. We would like to offer our sincere sympathies to his wife Ciara and his parents John and Rosemary and all the Delves family. The family have been my lifelong friends."
Mr Delves, a director of family printing business Masterphoto in Dublin 4, suffered severe head injuries in the incident, which police are treating as a tragic accident.
The apartment is located on Calle Tenerifein the small resort town of El Medano, which is in the south of Tenerife.
Mr Delves had celebrated his birthday last Sunday. Married to Ciara, he had three young children aged 12, four and two.
His brother, Colm, is group chief executive with telecoms company Digicel.
A Rush woman fought back her tears as she spoke of her memories of the businessman who was devoted to his family.
"Robert and his family did everything together. They were often seen out walking on the local roads together or he would be down on the beach playing with his children.
"He would arrive home and the children would be playing in the garden and they would welcome him. They're such a lovely family," she said.
Mr Delves was a keen sportsman and a respected coach at St Maurs GAA club in Rush.
"He was a very popular member of the club, and a very popular man locally," club vice-chair Owen McGinty said. "He was the most inoffensive guy you would ever meet and a very hard worker. It's a terrible loss."
Leo Bisset, a friend of Mr Delves, said the close-knit coastal town of Rush was shattered and stunned by his death.
"Robert was an absolute gentleman, one the nicest you could ever meet," he said. "He was a superb footballer and golfer. You would never, ever hear a bad word about Robert or his family, lovely people."
Another St Maurs club member said he was a "lovely man, very quiet and very gentle". He added he was involved in both the senior team, and in coaching young players.
A post mortem was due to be carried out last night at Tenerife's Institute of Forensic Medicine in the capital, Santa Cruz.
The Department of Foreign Affairs confirmed it was providing consular assistance through its embassy in Madrid.