The Light Blues have been looking for somebody to take the reins following the sacking of Pedro Caixinha but were refused permission to speak to McInnes after making an approach on Tuesday.
After McInnes spent two days "weighing up the opportunity", the Dons announced on Thursday that the 46-year-old had chosen to remain at Pittodrie.
An official Gers statement read: "The position of Rangers manager requires an ability to win football matches and the mentality to cope with the demanding off-pitch environment that goes with being the Rangers manager. This is a critical aspect of our assessment of any candidate during the interview process.
"We endorse [McInnes's] position because moving to a massive club like Rangers is a big step with concomitant risk. We continue to consider candidates but will only appoint someone in whom we have full confidence and who feels he is ready for the job.
"In the meantime, we have great confidence in Graeme Murty, who will continue as interim manager at least until the end of the year. The manner in which he has approached this task is a credit to himself and the club."
Murty, the club's Under-20s coach, has guided the side to four wins in six games in his second spell as caretaker manager.