In Star Trek: Voyager's very next episode, "Virtuoso", the Doctor discovers that an alien race values his singing talent, so he attempts to retire to their planet. If Voyager employed more serialized storytelling, this could be interpreted as the Doctor trying to recreate the life he had away from the USS Voyager in "Blink of an Eye".
As a hologram, the Doctor easily adapts to the time differential, which also—unfortunately—means the Doctor bounces back to life on Voyager relatively easily. The Doctor briefly reports that he built a whole new life while waiting for the mission's end, with a singing career, new friends, a long-term girlfriend, and even a son. This planet's people were clearly important to the Doctor, and they helped him in his overall character arc of becoming a person. Instead, Star Trek: Voyager's frequent return to status quo means the storytelling potential of the Doctor's extra three years goes unexplored.
Robert Picardo's USS Voyager Doctor Exists In Star Trek's 32nd Century
Will Star Trek: Starfleet Academy's Version Of The Doctor Have Centuries To Catch Up on?
The announcement of Star Trek legacy characters in Star Trek: Starfleet Academy reveals that Robert Picardo's USS Voyager Doctor exists in Star Trek's 32nd century. That doesn't necessarily mean that the Doctor has been actively operational for all 800 years between Star Trek: Voyager and Star Trek: Starfleet Academy. Starfleet Academy's Doctor could be a backup copy like the 31st-century version from Star Trek: Voyager season 4, episode 23, "Living Witness", but if not, the Doctor's ability to adjust easily to the planet's time differential in "Blink of an Eye" could point to another explanation.
As part of the USS Voyager-A crew in Star Trek: Prodigy, the Doctor would return to the Delta Quadrant to fix Voyager's mistakes. Voyager left the "Blink of an Eye" planet hoping they would learn to stabilize the time differential. The Doctor would still be the ideal crew member to beam down and check in. If the time-dilated society accidentally overshot stabilizing the differential so it became inverted in the intervening years, a brief mission for Star Trek: Voyager's Doctor would add up to far more than just ten years in the Delta Quadrant ... maybe even centuries.