Technically maybe not, Springsteen learned his self guitar playing, can also play piano, hammond, harmonica and drums. But he is for me the best songwriter and bandleader, The E Street band is the best live band. And looking at the number of songs he has, writen, shows of more than 3 hours, knows his texts and never the same setlist, you can say Springsteen is also great musician. And don't forget every month a official download of a Springsteen show and shows with The E street Band in 2014 and 2016 are all available.
Springsteen is foremost an excellent songwriter. His lyrics are without any doubt of literary quality and he is a great and very original composer of songs. He is also a magnificent band-leader. A Rock 'N Roll conductor, as it were. A very talented guy, so to say.
All of this doesn't answer the question of whether he is a good musician. It's obvious he can play the chords (on guitar and keyboard), which makes him a fine musician for sure. But his true musical performance strength is in his singing. Now I guess it depends on how "musical" you want to take the genre of Rock 'N Roll (or just "Rock"), but as a Rock singer, I think he is one of the best. He surely has the most singing stamina of all the greats (Jagger, Bono, Freddie, Elvis, Jim Morrison) and a particular rawness that is all natural at the same time.
Not talented enough myself to be able to comment on how good a musician he is but he is most certainly a rock and roll evangelist and a showman. I've seen a lot of artists over nearly 40 years of concert going and, after seeing him for the first time only 2 years ago, there is no doubt in my mind that he ranks first in terms of the energy and stamina needed to put on a 3 hour show. He leaves artists half his age in his wake, even our beloved U2 can't pull of shows of that duration.
If I can say anything about his musicianship, it is that I admire his ability to rework a song and make it sound completely different. That takes a bit of skill and it's worth watching or listening to his Broadway show to get a few examples of that.