In Back to the Future Part II, Doc tells a lot about the future of Marty's family. However, he doesn't mention what happened to himself in 2015. Was Doc supposed to be dead in 2015 (if he had never traveled to 1885)?
Doc's life after the events of BTTF II is pretty well recorded in the sequel BTTF 3. He was ejected into the 1880s where he worked as a blacksmith. He met and wooed his future wife and assisted Marty in returning to the 1980s. Some 8-10 years later he was successful in building a time-traveling train and he and his wife (and two children and dog) roamed throughout time and space having adventures.
There's no good indication that Doc Brown spent any time living in the 2010s although we know that he periodically visited in order to pick up hover-conversion technology and fusion power generation equipment.
As to whether he would have lived to 2015 had time travel not been invented, the answer is that we simply don't know. The game, novel and animated feature all give him a different date of birth but we know that he would have been approaching his hundredth birthday. There was certainly rejuvenation technology available in the 2000s, but it's quite reasonable to assume that he would have died before this became widespread.