No, the injury or illness has to happen before the second half of the season begins, and must be incapacitating for the rest of the season.
Hardship waivers (which is what most people mean by "medical redshirt") are not decided by the team or the school. Only the conference or the NCAA Committee on Student-Athlete Reinstatement can grant a waiver, and it is based on medical documentation. Obviously if a player does compete, he is capable of playing, but not playing in games in no way demonstrates that the player was incapacitated.