Unlike Titanfall, it is possible to play Destiny all by your lonesome, and ignore all the grouping and social features -- to a point. Bungie's Jason Jones explained exactly what they have planned to Game Informer, and it looks like you'll be able to enjoy most, but not all of the game solo.
Jones says that "You absolutely are going to be able to play Destiny by yourself and have the same kind of fun shooter-experience that you could have in a single-player campaign," but he goes on to also state, "I would say that there's some sort of - if you wanted to talk about it in MMO terms, you'd say 'end-game activities', but some of the most intense non-competitive activities in the game do require cooperation. They require a group of players to tackle at once."
This should come as no surprise to loot game fanatics, as many titles have a soft requirement for end-game raids, including Borderlands 2 and Diablo III's Inferno Mode. Of course, there are freaks of nature out there who can go at it solo regardless of the design, but for the vast majority of you, expect to do some grouping at some point.