Hi,
I had this exact same issue when I had the beta and was unable to play it. As soon as I press X on the main screen i get the following message "Attention: You must be connected to Xbox Live in order to play Destiny" I am connected to Xbox live and even ran all the tests, NAT is open. Fair enough its not the fastest (according to Xbox test 3mb down 0.6 up) but I only want to play it on single player.
Can someone please help?
Thank you
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What I found is that even though I was signed in and everything worked (well, except for Destiny) I still had to sign out of XBL and then back in. That resolved this problem for me.