i think that their primary connection was Neal Cassady. I'm sure that Garcia read Keroac, but I've never heard of him saying that he felt that any sort of torch had been passed to him from Kerouac. Garcia wasn't a lyricist, so there's no connection there. It's possible that Robert Hunter, who wrote most of Garcia's lyrics, felt some connection to Kerouac, but I don't know.
I recommend the official Grateful Dead history, Long Strange Trip, by their publicist Dennis McNally, who also happens to have written a book on Kerouac and the Beats (called "Desolate Angel")