Ms Gerard, 28, was last seen outside Irish bar Molly Malone's in Puerto de la Cruz in the early hours of November 30.
She worked at Tenerife's Loro Parque marine theme park and lived with boyfriend and fellow marine wildlife trainer Dennis Kissling.
The body was discovered in the sea opposite the lighthouse on Tuesday, December 4 in Puerto de la Cruz.
In a statement released by the Lucie Blackman Trust, the family said DNA results confirmed their "worst nightmare" and that they are "heartbroken".
The statement read: "The DNA results confirm our worst nightmare and we are all heartbroken that Amy is no longer with us.
"The police investigation is ongoing and there is no further information to share at this time."
The Lucie Blackman Trust has assisted thousands of families of missing people abroad.
Matthew Searle MBE, Lucie Blackman Trust Chief Executive, said: "This is the worst news for the family just before Christmas.
"The thoughts of all of us here at the charity are with Amy's family and we continue to support them.
"The family have asked for privacy and we hope this is afforded them."
According to Ms Gerard's Facebook profile, the University of Lincoln graduate, originally from Cleethorpes, trained dolphins and killer whales at the zoo, where she had been working since May 2016.