The 26-year-old was born in Slovenia and represented them in three Davis Cup ties before becoming a British citizen last March.
The ITF introduced a rule later in 2015 that players would not be allowed to play for a second country in the most prestigious team competition in men's tennis.
"Of course I am very sad with today's result and that it has not gone the way I had hoped," world number 57 Bedene said.
"I will have to assess my situation now before I decide on what steps to take next."
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Britain, led by Andy Murray, won the Davis Cup last year for the first time since 1936.
(Editing by Ed Osmond)
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