Eric Roberts Apologizes to Sister Julia in New Memoir

Eric Roberts is offering a mea culpa to his little sis, Julia Roberts, saying it wasn't cool for him to take credit for her career years ago ... and, he laments their fractured relationship.

People released several excerpts from the prolific actor's upcoming memoir, "Runaway Train" ... where the outlet says he writes a public apology to Julia after years of estrangement.

Back in the day, ER says Julia's fame quickly overtook his own ... and, he defensively told reporters there "would be no Julia Roberts" without him charting the path.

Looking back on it ... Eric calls the comment "asinine" -- and, adds he hopes JR will accept the apology.

Eric peels back the curtain on much of his life ... discussing his parents' difficult marriage -- when they split, he went with their dad, and Julia lived with their mom -- and the 1981 car crash that put him in a coma.

He also delves into his past drug addiction ... admitting more tension cropped up between himself and Julia during a custody battle over his daughter Emma Roberts with his ex, Kim Cunningham -- who Julia sided with. Eric was pissed about the move at the time, but writes he thinks it was the right call in hindsight.

Eric -- who is still estranged from his sister and his daughter -- says he knows he will go down in history as Julia's brother and Emma's dad ... and, he claims he's fine with that.