Hunter Biden's Divorce Will NOT Be Sealed from the Public

Hunter and Kathleen Biden's push to keep their divorce battle out of the public eye has been shot down by the judge ... TMZ has learned.

According to docs obtained by TMZ, Joe Biden's son and daughter-in-law claim they and their kids are public figures, and if details of their divorce get out ... it could be "damaging and embarrassing" to the family. Hence the request to seal the case.

The court called B.S., saying if the estranged couple wanted to keep it on the DL, they shouldn't have included the sordid allegations in the initial divorce petition, which was filed publicly. The judge says in docs, "Sealing those documents now will not erase any embarrassment the parties or their children may have already experienced."

You'll recall ... each accused the other of cheating, and she said Hunter blew all their money.