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Gwen Stefani and Blake Shelton are shutting down the rumor mill without saying a word ... stepping out for a grocery run and looking totally at ease together despite a wave of divorce rumors online.

The two were photographed Tuesday in Los Angeles while shopping at the upscale Italian market, Eataly ... grabbing groceries side by side before heading out with smiles that didn't leave much room for speculation. Backgrid

Sources close to the couple tell TMZ ... the talk is way off base, saying the pair is "completely baffled" by the divorce rumors. Our sources say they don't even know where the rumors came from, but that they're complete nonsense and they’re very happy together.

Our sources add the couple just celebrated Thanksgiving in Los Angeles with Gwen's family,

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