“As long as the vice of gluttony has a hold on a man, all that he has done valiantly is forfeited by him: and as long as the belly is unrestrained, all virtue comes to naught.” But virtue is not done away save by mortal sin. Therefore gluttony is a mortal sin. (Summa Theologiae, II-IIae, Q. 148, a. 1 & 2).
“Not that which goeth into the mouth defileth a man: but what cometh out of the mouth, this defileth a man.” (Jesus)
Eating with unwashed hands and gluttony are two different issues.