The Shrek and Charlie's Angels star, 35, also thanks those who have extended their condolences.
"My mom, sister and I are deeply appreciative of the overwhelming outpouring of love and support during this difficult time for us and all of our family," says the message from the actress, released by her representative, Brad Cafarelli.
"My dad was loved by so many people, and his humor and spirit will always live on in our hearts," she added.
Diaz's sister, Chimene, is 37. Their mother, the former Billie Early, is an import-export businesswoman.
Emilio Diaz, who was 58, reportedly worked for UNOCAL (formerly the Union Oil Company) as a field gauger in the pipeline department for more than 20 years, until his retirement in 1998.
On Tuesday, filming of Diaz's current film, My Sister's Keeper, was halted.
Reporting by JULIE JORDAN