When using a variable, you need to use double quotes. Otherwise, it will be expanded to a few words. On the other hand, it is not necessary to quote the value of a variable twice. The following should work:
header='--header=User-Agent: Mozilla/5.0 (Windows NT 6.0) AppleWebKit/537.11 (KHTML, like Gecko) Chrome/23.0.1271.97 Safari/537.11' wget "$header" http://website.com -O index
Edit: If you want to use variables for multiple arguments, use arrays:
args=(-d '--header=User-Agent: Mozilla/5.0 (Windows NT 6.0) AppleWebKit/537.11 (KHTML, like Gecko) Chrome/23.0.1271.97 Safari/537.11') wget "${args[@]}" http://website.com -O index
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