No, multiplayer is not always necessary. Multiplayer games should be designed from the ground up to be competitive and balanced, rather than story based and immersive. Single-player games, similarly, should be designed from the ground up to be the other way around.
In pretty much every genre you have games that favour one or the other. For example:
RPGS: MMORPGs vs Bethesda/Bioware/JRPGs
FPSs: Halo and MW for consoles, the Q3, CS, L4D, whatever-is-hot-now tradition for PC vs HL2, Bioshock, etc.
RTSs: This genre is more muddled, with a tendency towards more multiplayer and less story. One of the most critically acclaimed RTSs released recently, Sins of a Solar Empire, had no story at all, while for older series it was a must -> see Starcraft, AoE, and C&C.
Platformers/Adventure: Still singeplayer mostly, but that's just part of the inherent nature of the genre. That said, some are starting to implement competitive features e.g. Mirror's Edge's scoreboard system.
Sports: Match/Game/Race based vs strategic/managerial
That said, I worry about storybased FPSs being in decline. I miss stuff like NOLF, XIII and the Jedi Knight series...