Kind of close, but it was actually McMahon that called him into his office and told him things needed to change.
"The meeting where he asked to take on more responsibility- He felt the WWF product sucked and was hard to watch because of crap like TL Hopper and Who. He was watching the nWo angle around that time, so he started trying to bring them up to the times, as he felt that the company was stuck in 1986 while the rest of the world was in 1996. He went up to Linda and Vince McMahon and told them that he was thinking of going to WCW and that, if they thought he was only capable of writing the magazine, they should tell him that. They didn’t think that, so he stayed and the door was open for further opportunities. The next Monday, there was a memorably bad RAW where the matches were shot on one continent and the announcers were on another one. When the rating came in abysmally low, Vince McMahon had him brought to a big meeting and he was thinking “I’m in deep shit” until Vince threw a copy of the WWF magazine down on the table and said that the magazine was everything the show should be. It shocked him and probably wasn’t the best thing for him since Vince did it in that way, as it made people hate him. However, that did mean that he and McMahon were now writing the TV shows together."
The memorably bad Raw he's talking about seems to be April 14, 1997. http://rspw.org/petrie/970414.txt
After that, Russo started writing the shows with McMahon. What was Russo's first show? Why, it was this show. http://rspw.org/petrie/970421.txt
This could very well be the official start of the Attitude Era.