The song, which was unveiled yesterday, has been co-written by Smith with fellow Grammy-winner Jimmy Napes.
Smith, 23, becomes the first British solo male vocalist to sing the opening credits since Tom Jones in 1965.
Thanking his fans, the singer tweeted, "Reading through all your amazing comments. I'm so happy you like it and I can't believe it's still number one on iTunes!!"
Former Bond actor Roger Moore has called "Writing's On the Wall" "very haunting and wonderfully orchestrated".
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Other Twitter users have been less complimentary, with one describing it as the "worst Bond theme ever".
The song's title appears to derive from a scene in Spectre, viewed briefly in the film's trailer, in which Bond's name is found scrawled upon a wall.
"Spectre", which will feature Daniel Craig in his fourth performance as 007 agent, is due to be released in the UK simultaneously with its London world premiere on October 26 before hitting the screens in US on November 6 and India on November 20.