This question is based on a few misconceptions.
"Correct" is not a clearly defined concept for pronunciation. But so far as we can speak of "correct pronunciation" of English words, it's definitely not a matter of pronouncing things "as they are spelled". People generally consider it appropriate to use pronunciations that are widely used and not associated with particular stigmatized social groups, like lower-class speakers or speakers of stigmatized minority dialects.
There are many words that have a spelling that doesn't correspond to any widely used pronunciation. For example, the word friend is spelled with "ie", like the word fiend, but /friːnd/ ("freend") is not a widely used pronunciation in modern English (I have no idea if this pronunciation may exist in some obscure dialect), so it is not considered correct. The correct pronunciation of friend is instead considered to be /frɛnd/ ("frend"). So we can see that the idea of "correct pronunciation" cannot be based on spelling alone. There are other factors influencing what is considered "correct".
The pronunciation that would be expected for "bury" if it had a completely regular spelling would actually be /bjʊəri/, because there is only one R after the U (compare fury, jury and bureau).
But "bury" doesn't have a completely regular spelling. Rather, the spelling is based on weird details of its historical development. See the following note from the Etymonline entry for "bury":
The Old English -y- was a short "oo" sound, like modern French -u-. Under normal circumstances it transformed into Modern English -i- (in bridge, kiss, listen, sister, etc.), but in bury and a few other words (merry, knell) it retained a Kentish change to "e" that took place in the late Old English period. In the West Midlands, meanwhile, the Old English -y- sound persisted, slightly modified over time, giving the standard modern pronunciation of blush, much, church.
The people in the comments who say they have heard bury pronounced to rhyme with furry in an American accent may have heard someone with the with the "Merry–Murray merger". A speaker with this accent would also rhyme berry and ferry with furry.