The 20-year-old, who failed to progress beyond the first round of the Australian Open this month, has been suffering with the problem since last year.
"After consulting with my doctor I have been advised that I'll need to rest my wrist for another two to four weeks," she wrote on her Facebook page.
"This will sadly not allow me to play in Fed Cup, which was one of my biggest goals for 2014."
Robson will be replaced in the team by 22-year-old Jocelyn Rae.
"While we are obviously going to miss Laura on and off the court in Budapest, it's a great opportunity for me to bring Joss into the team as a doubles specialist," BBC Sport quotes GB captain Judy Murray as saying.
GB will face Hungary, Latvia and Romania in the group stage of the Fed Cup, which will take place between February 4 and 9.