Spock already knew that Kirk was acting 1st officer. Pike had designated/promoted him when they escorted him to the transport earlier.
Yes, but Spock neither gave command to Kirk, nor did he even acknowledge that he was suddenly OK with Kirk suddenly having command. Terribly out of character for Spock (no matter which universe we're talking about).
But Spock didn't have to specifically designate Kirk as the commanding officer. Spock knew that Kirk was 1st officer, and would assume command by right/responsibility, once he stepped down. That is what a 1st officer does if the captain cannot perform his/her duties.