While I’n not a big fan, John Mayer can play! And has had hits.
Bill Nelson would be my pick - not many hits - Ships In The Night, Maid in Heaven, Blazing Apostles all as Be-Bop De Luxe; but he is a virtuoso player.
MiM
I was wondering whether guitarists who were also known as songwriters (or singers, or band members, or producers) might have escaped your notice: you were less likely to think of them as “guitar gods” because they weren’t just guitarists. Adrian Belew, Brian May, and Lindsey Buckingham spring to mind as highly-regarded guitarists who are also great popular-style songwriters.
Stevie Ray Vaughn, maybe?
I started to add Billy Gibbons of ZZ Top and Ray Benson of Asleep at the Wheel, but I’m not so sure they fit the criteria.
I’ll second Mayer. I generally can’t stand his music, but the guy is a phenomenally talented guitarist and he has written some very popular, well-regarded songs.
Nah - not stupid, but the spectrum of Guitarist <----> Songwriter is broad and muddled, and where folks get placed on it is a huge debate in and of itself…
As I see the OP, the question is asking for excellent music writing and virtuoso guitar playing. Also that were big hits. Among our usual rock n roll suspects. I do not see that it is asking for the triple threat of lyricist. That said, here are my nominees:
Eric Clapton. Layla, Tears in Heaven, White Room, Sunshine of Your Love (last two co-written)
Jimi Hendrix. His “Little Wing” is a big hit with his fellow virtuosos, but not the public. He never went mainstream with Top 40 Hits.
Stevie Ray Vaughn. No big hits of his own.
David Gilmore. Comfortably Numb.
Jerry Garcia. Touch of Grey only Top 40 Hit.
George Harrison. Something, Here Comes the Sun, My Sweet Lord, While My Guitar Gently Weeps (really Clapton). I’ve some concerns about his being a “virtuoso”.
Duane Allman gets Ramblin Man, but did they have any other Top 40s?
If we add in lyrics, that narrows it down Clapton, Gilmore and Harrison. Require more than one hit, and you eliminate Gilmore. Clapton is unquestionably the better guitar player than Harrison (and may have played all the great guitar, so the prize goes to EC. Although EC cannot sing as well as Harrison, he is a passably good singer within his range.
Actually Layla was co-written with Jim Gordon, and Tears in Heaven was co-written with Will Jennings. The only hit song I could find that was written entirely by Clapton was Wonderful Tonight, as I said above.
Jim Gordon wrote the piano outro to Layla, so it was really Clapton’s songwriting that made it a hit. Though reports are the guitar riff that defines it was written by Duane Allman.
Jack White (listed by Rolling Stone as a guitar god) had at least one hit.
Mark Knopfler has a couple of hits; “Sultans of Swing” and “Money for Nothing” were great songs.
Dicky Betts wrote Ramblin Man after Duane died.
Here’s Clapton’s top singles, indicating US and UK chart positions. Promises is high up there, Wonderful Tonight. Hello Old Friend, I Can’t Stand It, (Another Ticket was not Top 40, but damn good lyrics)
Of those, Promises and I Can’t Stand It are my personal favorites. Clapton shares a lot of songwriting credits with people who never had another hit. Other than Harrison and Duane Allman, I’d consider that Clapton was the leading contributor based on his long history of participating in hits. With apologies to Jack Bruce and Ginger Baker, who were fine at what they did, but never did well without Clapton. Also special credit to Jim Gordon who did write the Layla piano code entirely himself and it is one of the great stretches of music of any kind.
Clapton considers many other guitarists to be better than he is, and I will agree with him that Hendrix and Vaughn were head and shoulders better, and Duane Allman in his lead in Layla too. But they were not as popular with the public and hit writing and all three died much too young. In my not expert opinion, Vaughn was technically the best of these, Hendrix the most amazing innovator and Allman the best and conveying his emotion through his virtuoso playing.
Good answer - very underrated.
I think your reasoning is flawed. Clapton earned his reputation and became a Big Name through his guitar. Once you become a Big Name, you get airplay and that makes it orders of magnitude easier to turn something with your Big Name on it into a hit. Consequently it just doesn’t follow that his long history of “participating” in hits means he was the leading contributor. It may well just mean that other no-name songwriters could get a hit with their songs if EC’s Big Name was associated with them.
I would say that what stands out about Clapton (in contrast to others as discussed below) is that a large proportion of his hits were either covers of other people’s songs (Sheriff, Cocaine, After Midnight) or collaborations with others as discussed above. Given that a Big Name like EC can use his name to launch his own songs, the fact that most of his hits are not or are not entirely his own strongly suggests that songwriting is not his strong suit. And I say this as a major EC fan.
If this OP is about writing popular songs (and it is) then consider someone like Knopfler. Pretty much every single track on all Dire Straits and Knopfler solo albums was written by him to the last note (control freak that his renowned for being).
Maybe in the U.S.; in the UK, “Purple Haze” reached no. 3, and “Voodoo Child” reached no. 1.
Taylor Swift, anyone? 
runs away
Lyle Lovett is mainly revered as a lyricist, but I think his guitar and vocals are up there too.
He is a solid player, but not a virtuoso. Vince Gill, Keith Urban, Brad Paisley - they are all big-time gunslingers in terms of technical precision - with Paisley at least able to hang with the metal shredders and Van Halen acrobatics…
No love for Dave Matthews? I think he’s under-appreciated as a guitarist.
As for Joni Mitchell… she’s a wonderful lyricists, but she’s basically a guitar strummer.
Peter Green is one of MY guitar gods, and while the Fleetwood Mac tunes “Black Magic Woman,” “Oh Well,” “Man of the World,” and “The Green Manalishi” may not have charted in the US, they were all hits in the UK. Green’s sole #1 hit in the UK, “Albatross,” was an instrumental, and so doesn’t fit the OP’s parameters.
Her work in alternate tunings is respected.
As for Dave, yeah, he’s a good guitar player and very successful songwriter. His stuff borders on riffy noodling, but can’t argue about his skills. I prefer John Mayer as a straight-up player who is popular these days…
Brian May wrote several of Queen’s hits, including “We Will Rock You”, “Tie Your Mother Down”, “Who Wants to Live Forever”, and “Fat Bottomed Girls”. Another of his compositions (though it was never a hit) is one of my favorite Queen songs, “'39”.