At the same time in California (where Les Paul spent some time) Paul Bigsby and Leo Fender were also experimenting. In 1947 Bigsby made a now famous solid body instrument with a single pickup for Merle Travis. It's infamous for looking like a cross between a "Les Paul" but with a headstock similar to a Strat. Obviously neither instrument existed, so that's not the case, (possibly the reverse) but is a clear indicator that cross pollinization was taking place.
In 1947 Leo formed the Fender Electric Instrument Company and in 1948 started to market the "Broadcaster" whose name was changed to the "Telecaster" in 1950. With the Broadcaster the modern solid body, two pickups, etc. had finally begun production.
The 40s also saw the birth of the modern, amplified pedal steel. Named the "Electraharp" and produced by Gibson, its history forks at this point because it really was no longer a guitar but something else entirely.
The electric bass guitar was first marketed by Fender in 1951, and it was this instrument that could be said to be his unique invention. I'd have to do more research, but the combination of bass range and guitar frets is probably his in a sense. My caveat is only because many early instruments had frets tied around the neck so the idea could have essentially been in place for hundreds of years at that point, but I believe he was the first to bring together bass range, frets and pickups into the first electric bass guitar.
As the market developed, Gibson produced the Les Paul in 1952 and Fender brought out the 3 pickup Stratocaster in 1953.
Remember that up until now the single coil pickup was all that was in production. (Think P90 and "soapbar" pickups on Gibsons at this point).
The next breakthrough was the work of Seth Lover and Walter Fuller who created the humbucking pickup in 1954. (It grew out of the needs of Radio broadcast and recording where, as to this day, quieter is better.) Gibson's ES 135 was the first to sport this innovation and entered into regular production on Les Paul guitars in late 1957. Early humbuckers have entered the mythology stamped with the now famous "patent applied for" phrase. The design of these pickups is exactly the same as is produced today.
At that point the essence of the modern electric guitar was in place. There have been a few innovations since. The Alembic company (formed in 1968 to supply the Grateful Dead's sound systems) were the first to put transistorized preamps in guitars and basses. Arnie Lazarus produced the FRAP (flat response audio pickup) piezo transducer in '69. Ovation may have been first to use individual tranducers under each string.