This is a courteous site. It would like to know where you come from what currency you use etc. to give you a better user experience. It does this by setting a multitude of cookies on the landing page. So we follow suit and navigate to the landing page first getting the cookies then we goto the page we want:

library(RCurl)
myURL <- "http://www.bluenile.com/api/public/loose-diamond/diamond-details/panel?country=USA&currency=USD&language=en-us&productSet=BN&sku=LD04077082"
agent="Mozilla/5.0 (Windows NT 6.3; WOW64; rv:32.0) Gecko/20100101 Firefox/32.0"

#Set RCurl pars
curl = getCurlHandle()
curlSetOpt(cookiejar="cookies.txt",  useragent = agent, followlocation = TRUE, curl=curl)
firstPage <- getURL("http://www.bluenile.com", curl=curl)
myPage <- getURL(myURL, curl = curl)

library(RJSONIO)
> names(fromJSON(myPage))
[1] "diamondDetailsHeader" "diamondDetailsBodies" "pageMetadata"         "expandedUrl"         
[5] "newVersion"           "multiDiamond"  

and the cookies:

> getCurlInfo(curl)$cookielist
 [1] ".bluenile.com\tTRUE\t/\tFALSE\t2412270275\tGUID\tDA5C11F5_E468_46B5_B4E8_D551D4D6EA4D"                                                                    
 [2] ".bluenile.com\tTRUE\t/\tFALSE\t1475342275\tsplit\tver~3&presetFilters~TEST"                                                                               
 [3] ".bluenile.com\tTRUE\t/\tFALSE\t1727630275\tsitetrack\tver~2&jse~0"                                                                                        
 [4] ".bluenile.com\tTRUE\t/\tFALSE\t1425230275\tpop\tver~2&china~false&french~false&ie~false&internationalSelect~false&iphoneApp~false&survey~false&uae~false" 
 [5] ".bluenile.com\tTRUE\t/\tFALSE\t1475342275\tdsearch\tver~6&newUser~true"                                                                                   
 [6] ".bluenile.com\tTRUE\t/\tFALSE\t1443806275\tlocale\tver~1&country~IRL&currency~EUR&language~en-gb&productSet~BNUK"                                         
 [7] ".bluenile.com\tTRUE\t/\tFALSE\t0\tbnses\tver~1&ace~false&isbml~false&fbcs~false&ss~0&mbpop~false&sswpu~false&deo~false"                                   
 [8] ".bluenile.com\tTRUE\t/\tFALSE\t1727630275\tbnper\tver~5&NIB~0&DM~-&GUID~DA5C11F5_E468_46B5_B4E8_D551D4D6EA4D&SESS-CT~1&STC~32RPVK&FB_MINI~false&SUB~false"
 [9] "#HttpOnly_www.bluenile.com\tFALSE\t/\tFALSE\t0\tJSESSIONID\tB8475C3AEC08205E5AC6252C94E4B858"                                                             
[10] ".bluenile.com\tTRUE\t/\tFALSE\t1727630278\tmigrationstatus\tver~1&redirected~false"