The 'Rocky' star spent his Friday touring Vatican City, and was actually granted an audience with Pope Francis, who went full fanboy ... telling Sly it's an honor to meet him because he grew up watching the actor's movies.

The video of their interaction is awesome -- Sly was bold enough to shadowbox the Pope, who even threw a few left hooks himself. Seriously, it's almost like the Pope is more thrilled about the meeting than Stallone.

The whole thing was a family affair ... Sly also introduced the Pope to his wife, Jennifer Flavin, and their three daughters, Sophia, Scarlet and Sistine. You'd think the pontiff might have been impressed by Sistine's name, but was more focused on the movie star of his younger years.

Sly went to the Vatican last September too, when he got to hold the keys that open every door in Vatican City, including the Sistine Chapel, and this time he got to meet His Holiness.