Today during Arsenal vs Crystal Palace, Pierre Emerick Aubamayeng's boot came out after a tackle and he couldn't retrieve the ball, but in case if he managed to retrieve the ball, is he allowed to continue to play immediately, without the boot?
Yes he can, but only until the ball next goes out of play.
According to IFAB Laws of the Game 2019/20 Law 04 Page 58
A player whose footwear or shinguard is lost accidentally must replace it as soon as possible and no later than when the ball next goes out of play; if before doing so the player plays the ball and/or scores a goal, the goal is awarded.