Crosbie suffered a shoulder injury and Gray is ruled out with a bicep issue.
Scotland claimed a dramatic first win in Cardiff for 22 years after withstanding a remarkable fightback from Wales to seal a 27-26 victory on Saturday.
Winger Darcy Graham and prop WP Nel are recovering well from their injuries and will both resume training with Edinburgh this week, Scottish Rugby's statement added.
Scotland face France in their second game on Saturday.